Interacting random systems

B. Jahnel, W. König, Probabilistic Methods in Telecommunications, D. Mazlum, ed., Compact Textbooks in Mathematics, Springer/Birkhäuser, Birkhäuser Basel, 2020, XI, 200 pages, (Monograph Published), DOI 10.1007/978-3-030-36090-0 .
Abstract
This textbook series presents concise introductions to current topics in mathematics and mainly addresses advanced undergraduates and master students. The concept is to offer small books covering subject matter equivalent to 2- or 3-hour lectures or seminars which are also suitable for self-study. The books provide students and teachers with new perspectives and novel approaches. They may feature examples and exercises to illustrate key concepts and applications of the theoretical contents. The series also includes textbooks specifically speaking to the needs of students from other disciplines such as physics, computer science, engineering, life sciences, finance.

W. König, Große Abweichungen, Techniken und Anwendungen, M. Brokate, A. Heinze , K.-H. Hoffmann , M. Kang , G. Götz , M. Kerz , S. Otmar, eds., Mathematik Kompakt, Springer, 2020, VIII, 167 pages, (Monograph Published), DOI 10.1007/978-3-030-52778-5 .
Abstract
Die Lehrbuchreihe Mathematik Kompakt ist eine Reaktion auf die Umstellung der Diplomstudiengänge in Mathematik zu Bachelor- und Masterabschlüssen. Inhaltlich werden unter Berücksichtigung der neuen Studienstrukturen die aktuellen Entwicklungen des Faches aufgegriffen und kompakt dargestellt. Die modular aufgebaute Reihe richtet sich an Dozenten und ihre Studierenden in Bachelor- und Masterstudiengängen und alle, die einen kompakten Einstieg in aktuelle Themenfelder der Mathematik suchen. Zahlreiche Beispiele und Übungsaufgaben stehen zur Verfügung, um die Anwendung der Inhalte zu veranschaulichen. Kompakt: relevantes Wissen auf 150 Seiten Lernen leicht gemacht: Beispiele und Übungsaufgaben veranschaulichen die Anwendung der Inhalte Praktisch für Dozenten: jeder Band dient als Vorlage für eine 2-stündige Lehrveranstaltung

P. Friz, W. König, Ch. Mukherjee, S. Olla, eds., Probability and Analysis in Interacting Physical Systems. In Honor of S.R.S. Varadhan, Berlin, August, 2016, 283 of Springer Proceedings in Mathematics & Statistics, Springer International Publishing, Cham, 2019, 294 pages, (Collection Published), DOI https://doi.org/10.1007/978-3-030-15338-0 .

W. König, The Parabolic Anderson Model -- Random Walks in Random Potential, Pathways in Mathematics, Birkhäuser, Basel, 2016, xi+192 pages, (Monograph Published).

W. König, ed., Mathematics and Society, European Mathematical Society Publishing House, Zurich, 2016, 314 pages, (Collection Published).
Abstract
The ubiquity and importance of mathematics in our complex society is generally not in doubt. However, even a scientifically interested layman would be hard pressed to point out aspects of our society where contemporary mathematical research is essential. Most popular examples are f inance, engineering, wheather and industry, but the way mathematics comes into play is widely unknown in the public. And who thinks of application fields like biology, encryption, architecture, or voting systems? This volume comprises a number of success stories of mathematics in our society ? important areas being shaped by cutting edge mathematical research. The authors are eminent mathematicians with a high sense for public presentation, addressing scientifically interested laymen as well as professionals in mathematics and its application disciplines.

P. Exner, W. König, H. Neidhardt, eds., Mathematical Results in Quantum Mechanics. Proceedings of the QMath12 Conference, World Scientific Publishing, Singapore, 2015, xii+383 pages, (Collection Published).

J.-D. Deuschel, B. Gentz, W. König, M. VON Renesse, M. Scheutzow, U. Schmock, eds., Probability in Complex Physical Systems, in Honour of Erwin Bolthausen and Jürgen Gärtner, 11 of Springer Proceedings in Mathematics, Springer, Berlin Heidelberg, 2012, 512 pages, (Collection Published).

S. Rjasanow, W. Wagner, Stochastic Numerics for the Boltzmann Equation, 37 of Springer Series in Computational Mathematics, Springer, Berlin, 2005, xiii+256 pages, (Monograph Published).

A. Hinsen, B. Jahnel, E. Cali, J.-P. Wary, Phase transitions for chase-escape models on Poisson--Gilbert graphs, Electronic Communications in Probability, 25 (2020), pp. 25/1--25/14, DOI 10.1214/20-ECP306 .
Abstract
We present results on phase transitions of local and global survival in a two-species model on Gilbert graphs. At initial time there is an infection at the origin that propagates on the Gilbert graph according to a continuous-time nearest-neighbor interacting particle system. The Gilbert graph consists of susceptible nodes and nodes of a second type, which we call white knights. The infection can spread on susceptible nodes without restriction. If the infection reaches a white knight, this white knight starts to spread on the set of infected nodes according to the same mechanism, with a potentially different rate, giving rise to a competition of chase and escape. We show well-definedness of the model, isolate regimes of global survival and extinction of the infection and present estimates on local survival. The proofs rest on comparisons to the process on trees, percolation arguments and finite-degree approximations of the underlying random graphs.

D.R.M. Renger, J. Zimmer, Orthogonality of fluxes in general nonlinear reaction networks, Discrete and Continuous Dynamical Systems -- Series S, published online on 19.05.2020, DOI 10.3934/dcdss.2020346 .
Abstract
We consider the chemical reaction networks and study currents in these systems. Reviewing recent decomposition of rate functionals from large deviation theory for Markov processes, we adapt these results for reaction networks. In particular, we state a suitable generalisation of orthogonality of forces in these systems, and derive an inequality that bounds the free energy loss and Fisher information by the rate functional.

V. Betz, H. Schäfer, L. Taggi, Interacting self-avoiding polygons, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 56 (2020), pp. 1321--1335, DOI 10.1214/19-AIHP1003 .
Abstract
We consider a system of self-avoiding polygons interacting through a potential that penalizes or rewards the number of mutual touchings and we provide an exact computation of the critical curve separating a regime of long polygons from a regime of localized polygons. Moreover, we prove the existence of a sub-region of the phase diagram where the self-avoiding polygons are space filling and we provide a non-trivial characterization of the regime where the polygon length admits uniformly bounded exponential moments

CH. Hirsch, B. Jahnel, A. Tóbiás, Lower large deviations for geometric functionals, Electronic Communications in Probability, 25 (2020), pp. 41/1--41/12, DOI 10.1214/20-ECP322 .
Abstract
This work develops a methodology for analyzing large-deviation lower tails associated with geometric functionals computed on a homogeneous Poisson point process. The technique applies to characteristics expressed in terms of stabilizing score functions exhibiting suitable monotonicity properties. We apply our results to clique counts in the random geometric graph, intrinsic volumes of Poisson--Voronoi cells, as well as power-weighted edge lengths in the random geometric, κ-nearest neighbor and relative neighborhood graph.

CH. Kwofie, I. Akoto, K. Opoku-Ameyaw, Modelling the dependency between inflation and exchange rate using copula, Journal of Probability and Statistics, 2020 (2020), DOI doi.org/10.1155/2020/2345746 .
Abstract
n this paper, we propose a copula approach in measuring the dependency between inflation and exchange rate. In unveiling this dependency, we first estimated the best GARCH model for the two variables. Then, we derived the marginal distributions of the standardised residuals from the GARCH. The Laplace and generalised t distributions best modelled the residuals of the GARCH(1,1) models, respectively, for inflation and exchange rate. These marginals were then used to transform the standardised residuals into uniform random variables on a unit interval [0, 1] for estimating the copulas. Our results show that the dependency between inflation and exchange rate in Ghana is approximately 7%.

B. Lees, L. Taggi, Site monotonicity and uniform positivity for interacting random walks and the spin O(N) model with arbitrary N, Communications in Mathematical Physics, 376 (2020), pp. 487--520, DOI https://doi.org/10.1007/s00220-019-03647-6 .

A. Tóbiás, B. Jahnel, Exponential moments for planar tessellations, Journal of Statistical Physics, 179 (2020), pp. 90--109, DOI 10.1007/s10955-020-02521-3 .
Abstract
In this paper we show existence of all exponential moments for the total edge length in a unit disc for a family of planar tessellations based on Poisson point processes. Apart from classical such tessellations like the Poisson--Voronoi, Poisson--Delaunay and Poisson line tessellation, we also treat the Johnson--Mehl tessellation, Manhattan grids, nested versions and Palm versions. As part of our proofs, for some planar tessellations, we also derive existence of exponential moments for the number of cells and the number of edges intersecting the unit disk.

F. Flegel, M. Heida, M. Slowik, Homogenization theory for the random conductance model with degenerate ergodic weights and unbounded-range jumps, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 55 (2019), pp. 1226--1257, DOI 10.1214/18-AIHP917 .
Abstract
We study homogenization properties of the discrete Laplace operator with random conductances on a large domain in Zd. More precisely, we prove almost-sure homogenization of the discrete Poisson equation and of the top of the Dirichlet spectrum. We assume that the conductances are stationary, ergodic and nearest-neighbor conductances are positive. In contrast to earlier results, we do not require uniform ellipticity but certain integrability conditions on the lower and upper tails of the conductances. We further allow jumps of arbitrary length. Without the long-range connections, the integrability condition on the lower tail is optimal for spectral homogenization. It coincides with a necessary condition for the validity of a local central limit theorem for the random walk among random conductances. As an application of spectral homogenization, we prove a quenched large deviation principle for thenormalized and rescaled local times of the random walk in a growing box. Our proofs are based on a compactness result for the Laplacian's Dirichlet energy, Poincaré inequalities, Moser iteration and two-scale convergence

F. Flegel, M. Heida, The fractional p-Laplacian emerging from homogenization of the random conductance model with degenerate ergodic weights and unbounded-range jumps, Calculus of Variations and Partial Differential Equations, 59 (2020), pp. 8/1--8/39 (published online on 28.11.2019), DOI 10.1007/s00526-019-1663-4 .
Abstract
We study a general class of discrete p-Laplace operators in the random conductance model with long-range jumps and ergodic weights. Using a variational formulation of the problem, we show that under the assumption of bounded first moments and a suitable lower moment condition on the weights, the homogenized limit operator is a fractional p-Laplace operator. Under strengthened lower moment conditions, we can apply our insights also to the spectral homogenization of the discrete Lapalace operator to the continuous fractional Laplace operator.

V. Betz, L. Taggi, Scaling limit of ballistic self-avoiding walk interacting with spatial random permutations, Electronic Journal of Probability, 24 (2019), pp. 74/1--74/33, DOI 10.1214/19-EJP328 .

CH. Hirsch, B. Jahnel, Large deviations for the capacity in dynamic spatial relay networks, Markov Processes and Related Fields, 25 (2019), pp. 33--73.
Abstract
We derive a large deviation principle for the space-time evolution of users in a relay network that are unable to connect due to capacity constraints. The users are distributed according to a Poisson point process with increasing intensity in a bounded domain, whereas the relays are positioned deterministically with given limiting density. The preceding work on capacity for relay networks by the authors describes the highly simplified setting where users can only enter but not leave the system. In the present manuscript we study the more realistic situation where users leave the system after a random transmission time. For this we extend the point process techniques developed in the preceding work thereby showing that they are not limited to settings with strong monotonicity properties.

B. Lees, L. Taggi, Site monotonicity and uniform positivity for interacting random walks and the spin O(N) model with arbitrary N, Communications in Mathematical Physics, pp. published online on 07.12.2019, urlhttps://doi.org/10.1007/s00220-019-03647-6, DOI 10.1007/s00220-019-03647-6 .

A.D. Mcguire, S. Mosbach, G. Reynolds, R.I.A. Patterson, E.J. Bringley, N.A. Eaves, J. Dreyer, M. Kraft, Analysing the effect of screw configuration using a stochastic twin-screw granulation model, , 203 (2019), pp. 358--379, DOI https://doi.org/10.1016/j.ces.2019.03.078 .

L. Andreis, A. Asselah, P. Dai Pra , Ergodicity of a system of interacting random walks with asymmetric interaction, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 55 (2019), pp. 590--606.
Abstract
We study N interacting random walks on the positive integers. Each particle has drift delta towards infinity, a reflection at the origin, and a drift towards particles with lower positions. This inhomogeneous mean field system is shown to be ergodic only when the interaction is strong enough. We focus on this latter regime, and point out the effect of piles of particles, a phenomenon absent in models of interacting diffusion in continuous space.

B. Jahnel, Ch. Külske, Attractor properties for irreversible and reversible interacting particle systems, Communications in Mathematical Physics, 366 (2019), pp. 139--172, DOI 10.1007/s00220-019-03352-4 .
Abstract
We consider translation-invariant interacting particle systems on the lattice with finite local state space admitting at least one Gibbs measure as a time-stationary measure. The dynamics can be irreversible but should satisfy some mild non-degeneracy conditions. We prove that weak limit points of any trajectory of translation-invariant measures, satisfying a non-nullness condition, are Gibbs states for the same specification as the time-stationary measure. This is done under the additional assumption that zero entropy loss of the limiting measure w.r.t. the time-stationary measure implies that they are Gibbs measures for the same specification.We also give an alternate version of the last condition such that the non-nullness requirement can be dropped. For dynamics admitting a reversible Gibbs measure the alternative condition can be verified, which yields the attractor property for such dynamics. This generalizes convergence results using relative entropy techniques to a large class of dynamics including irreversible and non-ergodic ones. We use this to show synchronization for the rotation dynamics exhibited in citeJaKu12 possibly at low temperature, and possibly non-reversible. We assume the additional regularity properties on the dynamics: 1 There is at least one stationary measure which is a Gibbs measure. 2 Zero loss of relative entropy density under dynamics implies the Gibbs property.

B. Jahnel, Ch. Külske, Gibbsian representation for point processes via hyperedge potentials, Journal of Theoretical Probability, pp. published online on 03.11.2019, urlhttps://doi.org/10.1007/s10959-019-00960-7, DOI 10.1007/s10959-019-00960-7 .
Abstract
We consider marked point processes on the d-dimensional euclidean space, defined in terms of a quasilocal specification based on marked Poisson point processes. We investigate the possibility of constructing uniformly absolutely convergent Hamiltonians in terms of hyperedge potentials in the sense of Georgii [2]. These potentials are natural generalizations of physical multibody potentials which are useful in models of stochastic geometry.

W. König, A. Tóbiás, A Gibbsian model for message routeing in highly dense multihop networks, ALEA. Latin American Journal of Probability and Mathematical Statistics, 16 (2019), pp. 211--258, DOI 10.30757/ALEA.v16-08 .
Abstract
We investigate a probabilistic model for routing in relay-augmented multihop ad-hoc communication networks, where each user sends one message to the base station. Given the (random) user locations, we weigh the family of random, uniformly distributed message trajectories by an exponential probability weight, favouring trajectories with low interference (measured in terms of signal-to-interference ratio) and trajectory families with little congestion (measured by how many pairs of hops use the same relay). Under the resulting Gibbs measure, the system targets the best compromise between entropy, interference and congestion for a common welfare, instead of a selfish optimization. We describe the joint routing strategy in terms of the empirical measure of all message trajectories. In the limit of high spatial density of users, we derive the limiting free energy and analyze the optimal strategy, given as the minimizer(s) of a characteristic variational formula. Interestingly, expressing the congestion term requires introducing an additional empirical measure.

W. König, A. Tóbiás, Routeing properties in a Gibbsian model for highly dense multihop networks, IEEE Transactions on Information Theory, 65 (2019), pp. 6875--6897, DOI 10.1109/TIT.2019.2924187 .
Abstract
We investigate a probabilistic model for routeing in a multihop ad-hoc communication network, where each user sends a message to the base station. Messages travel in hops via the other users, used as relays. Their trajectories are chosen at random according to a Gibbs distribution that favours trajectories with low interference, measured in terms of sum of the signal-to-interference ratios for all the hops, and collections of trajectories with little total congestion, measured in terms of the number of pairs of hops arriving at each relay. This model was introduced in our earlier paper [KT17], where we expressed, in the high-density limit, the distribution of the optimal trajectories as the minimizer of a characteristic variational formula. In the present work, in the special case in which congestion is not penalized, we derive qualitative properties of this minimizer. We encounter and quantify emerging typical pictures in analytic terms in three extreme regimes. We analyze the typical number of hops and the typical length of a hop, and the deviation of the trajectory from the straight line in two regimes, (1) in the limit of a large communication area and large distances, and (2) in the limit of a strong interference weight. In both regimes, the typical trajectory turns out to quickly approach a straight line, in regime (1) with equally-sized hops. Surprisingly, in regime (1), the typical length of a hop diverges logarithmically as the distance of the transmitter to the base station diverges. We further analyze the local and global repulsive effect of (3) a densely populated area on the trajectories. Our findings are illustrated by numerical examples. We also discuss a game-theoretic relation of our Gibbsian model with a joint optimization of message trajectories opposite to a selfish optimization, in case congestion is also penalized

R.I.A. Patterson, D.R.M. Renger, Large deviations of jump process fluxes, Mathematical Physics, Analysis and Geometry. An International Journal Devoted to the Theory and Applications of Analysis and Geometry to Physics, 22 (2019), pp. 21/1--21/32, DOI 10.1007/s11040-019-9318-4 .
Abstract
We study a system of interacting particles that randomly react to form new particles. The reaction flux is the rescaled number of reactions that take place in a time interval. We prove a dynamic large-deviation principle for the reaction fluxes under general assumptions that include mass-action kinetics. This result immediately implies the dynamic large deviations for the empirical concentration.

D.R.M. Renger, Flux large deviations of independent and reacting particle systems, with implications for macroscopic fluctuation theory, Journal of Statistical Physics, 172 (2018), pp. 1291--1326, DOI 10.1007/s10955-018-2083-0 .
Abstract
We consider a system of independent particles on a finite state space, and prove a dynamic large-deviation principle for the empirical measure-empirical flux pair, taking the specific fluxes rather than net fluxes into account. We prove the large deviations under deterministic initial conditions, and under random initial conditions satisfying a large-deviation principle. We then show how to use this result to generalise a number of principles from Macroscopic Fluctuation Theory to the finite-space setting.

D.R.M. Renger, P. Koltai , From large deviations to transport semidistances: Coherence analysis for finite Lagrangian data, Journal of Non-Newtonian Fluid Mechanics, 28 (2018), pp. 1915--1957, DOI 10.1007/s00332-018-9471-0 .
Abstract
Onsager's 1931 `reciprocity relations' result connects microscopic time-reversibility with a symmetry property of corresponding macroscopic evolution equations. Among the many consequences is a variational characterization of the macroscopic evolution equation as a gradient-flow, steepest-ascent, or maximal-entropy-production equation. Onsager's original theorem is limited to close-to-equilibrium situations, with a Gaussian invariant measure and a linear macroscopic evolution. In this paper we generalize this result beyond these limitations, and show how the microscopic time-reversibility leads to natural generalized symmetry conditions, which take the form of generalized gradient flows.

D.R.M. Renger, Gradient and GENERIC systems in the space of fluxes, applied to reacting particle systems, Entropy. An International and Interdisciplinary Journal of Entropy and Information Studies, 20 (2018), pp. 596/1--596/26, DOI 10.3390/e20080596 .
Abstract
In a previous work we devised a framework to derive generalised gradient systems for an evolution equation from the large deviations of an underlying microscopic system, in the spirit of the Onsager-Machlup relations. Of particular interest is the case where the microscopic system consists of random particles, and the macroscopic quantity is the empirical measure or concentration. In this work we take the particle flux as the macroscopic quantity, which is related to the concentration via a continuity equation. By a similar argument the large deviations can induce a generalised gradient or Generic system in the space of fluxes. In a general setting we study how flux gradient or generic systems are related to gradient systems of concentrations. The arguments are explained by the example of reacting particle systems, which is later expanded to include spatial diffusion as well.

C. Cotar, B. Jahnel, Ch. Külske, Extremal decomposition for random Gibbs measures: From general metastates to metastates on extremal random Gibbs measures, Electronic Communications in Probability, 23 (2018), pp. 1--12, DOI 10.1214/18-ECP200 .
Abstract
The concept of metastate measures on the states of a random spin system was introduced to be able to treat the large-volume asymptotics for complex quenched random systems, like spin glasses, which may exhibit chaotic volume dependence in the strong-coupling regime. We consider the general issue of the extremal decomposition for Gibbsian specifications which depend measurably on a parameter that may describe a whole random environment in the infinite volume. Given a random Gibbs measure, as a measurable map from the environment space, we prove measurability of its decomposition measure on pure states at fixed environment, with respect to the environment. As a general corollary we obtain that, for any metastate, there is an associated decomposition metastate, which is supported on the extremes for almost all environments, and which has the same barycenter.

M. Biskup, R. Fukushima, W. König, Eigenvalue fluctuations for lattice Anderson Hamiltonians: Unbounded potentials, Interdisciplinary Information Sciences, 24 (2018), pp. 59--76.
Abstract
We consider random Schrödinger operators with Dirichlet boundary conditions outside lattice approximations of a smooth Euclidean domain and study the behavior of its lowest-lying eigenvalues in the limit when the lattice spacing tends to zero. Under a suitable moment assumption on the random potential and regularity of the spatial dependence of its mean, we prove that the eigenvalues of the random operator converge to those of a deterministic Schrödinger operator. Assuming also regularity of the variance, the fluctuation of the random eigenvalues around their mean are shown to obey a multivariate central limit theorem. This extends the authors' recent work where similar conclusions have been obtained for bounded random potentials. endabstract

G. Botirov, B. Jahnel, Phase transitions for a model with uncountable spin space on the Cayley tree: The general case, Positivity. An International Mathematics Journal Devoted to Theory and Applications of Positivity, 23 (2019), pp. 291--301 (published online on 17.08.2018), DOI 10.1007/s11117-018-0606-1 .
Abstract
In this paper we complete the analysis of a statistical mechanics model on Cayley trees of any degree, started in [EsHaRo12, EsRo10, BoEsRo13, JaKuBo14, Bo17]. The potential is of nearest-neighbor type and the local state space is compact but uncountable. Based on the system parameters we prove existence of a critical value θ _c such that for θ≤θ _c there is a unique translation-invariant splitting Gibbs measure. For θ _c < θ there is a phase transition with exactly three translation-invariant splitting Gibbs measures. The proof rests on an analysis of fixed points of an associated non-linear Hammerstein integral operator for the boundary laws.

CH. Hirsch, B. Jahnel, E. Cali, Continuum percolation for Cox point processes, Stochastic Processes and their Applications, 366 (2019), pp. 139--172 (published online on 20.11.2018), DOI 10.1016/j.spa.2018.11.002 .
Abstract
We investigate continuum percolation for Cox point processes, that is, Poisson point processes driven by random intensity measures. First, we derive sufficient conditions for the existence of non-trivial sub- and super-critical percolation regimes based on the notion of stabilization. Second, we give asymptotic expressions for the percolation probability in large-radius, high-density and coupled regimes. In some regimes, we find universality, whereas in others, a sensitive dependence on the underlying random intensity measure survives.

L. Andreis, P. Dai Pra, M. Fischer, McKean--Vlasov limit for interacting systems with simultaneous jumps, Stochastic Analysis and Applications, 36 (2018), pp. 960--995, DOI 10.1080/07362994.2018.1486202 .
Abstract
Motivated by several applications, including neuronal models, we consider the McKean-Vlasov limit for mean-field systems of interacting diffusions with simultaneous jumps. We prove propagation of chaos via a coupling technique that involves an intermediate process and that gives a rate of convergence for the W1 Wasserstein distance between the empirical measures of the two systems on the space of trajectories D([0,T],R^d).

F. Flegel, Localization of the principal Dirichlet eigenvector in the heavy-tailed random conductance model, Electronic Journal of Probability, 23 (2018), pp. 68/1--68/43, DOI doi:10.1214/18-EJP160 .
Abstract
We study the asymptotic behavior of the principal eigenvector and eigenvalue of the random conductance Laplacian in a large domain of Zd (d ≥ 2) with zero Dirichlet condition. We assume that the conductances w are positive i.i.d. random variables, which fulfill certain regularity assumptions near zero. If γ = sup q ≥ 0; E [w^-q]<∞ <¼, then we show that for almost every environment the principal Dirichlet eigenvector asymptotically concentrates in a single site and the corresponding eigenvalue scales subdiffusively. The threshold γrm c = ¼ is sharp. Indeed, other recent results imply that for γ>¼ the top of the Dirichlet spectrum homogenizes. Our proofs are based on a spatial extreme value analysis of the local speed measure, Borel-Cantelli arguments, the Rayleigh-Ritz formula, results from percolation theory, and path arguments.

W. Wagner, A random walk model for the Schrödinger equation, Mathematics and Computers in Simulation, 143 (2018), pp. 138--148, DOI 10.1016/j.matcom.2016.07.012 .
Abstract
A random walk model for the spatially discretized time-dependent Schrödinger equation is constructed. The model consists of a class of piecewise deterministic Markov processes. The states of the processes are characterized by a position and a complex-valued weight. Jumps occur both on the spatial grid and in the space of weights. Between the jumps, the weights change according to deterministic rules. The main result is that certain functionals of the processes satisfy the Schrödinger equation.

M. Biskup, W. König, R. Soares Dos Santos, Mass concentration and aging in the parabolic Anderson model with doubly-exponential tails, Probability Theory and Related Fields, 171 (2018), pp. 251--331 (published online on 27.05.2017), DOI 10.1007/s00440-017-0777-x .
Abstract
We study the solutions to the Cauchy problem on the with random potential and localised initial data. Here we consider the random Schr?dinger operator, i.e., the Laplace operator with random field, whose upper tails are doubly exponentially distributed in our case. We prove that, for large times and with large probability, a majority of the total mass of the solution resides in a bounded neighborhood of a site that achieves an optimal compromise between the local Dirichlet eigenvalue of the Anderson Hamiltonian and the distance to the origin. The processes of mass concentration and the rescaled total mass are shown to converge in distribution under suitable scaling of space and time. Aging results are also established. The proof uses the characterization of eigenvalue order statistics for the random Schrödinger operator in large sets recently proved by the first two authors.

E. Bolthausen, W. König, Ch. Mukherjee, Mean-field interaction of Brownian occupation measures. II: A rigorous construction of the Pekar process, Communications on Pure and Applied Mathematics, 70 (2017), pp. 1598--1629.
Abstract
We consider mean-field interactions corresponding to Gibbs measures on interacting Brownian paths in three dimensions. The interaction is self-attractive and is given by a singular Coulomb potential. The logarithmic asymptotics of the partition function for this model were identified in the 1980s by Donsker and Varadhan [DV83] in terms of the Pekar variational formula, which coincides with the behavior of the partition function corresponding to the polaron problem under strong coupling. Based on this, Spohn ([Sp87]) made a heuristic observation that the strong coupling behavior of the polaron path measure, on certain time scales, should resemble a process, named as the itPekar process, whose distribution could somehow be guessed from the limiting asymptotic behavior of the mean-field measures under interest, whose rigorous analysis remained open. The present paper is devoted to a precise analysis of these mean-field path measures and convergence of the normalized occupation measures towards an explicit mixture of the maximizers of the Pekar variational problem. This leads to a rigorous construction of the aforementioned Pekar process and hence, is a contribution to the understanding of the “mean-field approximation" of the polaron problem on the level of path measures. The method of our proof is based on the compact large deviation theory developed in [MV14], its extension to the uniform strong metric for the singular Coulomb interaction carried out in [KM15], as well as an idea inspired by a itpartial path exchange argument appearing in [BS97]

J.-D. Deuschel, P. Friz, M. Maurelli, M. Slowik, The enhanced Sanov theorem and propagation of chaos, Stochastic Processes and their Applications, 128 (2018), pp. 2228--2269 (published online on 21.09.2017), DOI 10.1016/j.spa.2017.09.010 .
Abstract
We establish a Sanov type large deviation principle for an ensemble of interacting Brownian rough paths. As application a large deviations for the (k-layer, enhanced) empirical measure of weakly interacting diffusions is obtained. This in turn implies a propagation of chaos result in a space of rough paths and allows for a robust analysis of the particle system and its McKean?Vlasov type limit, as shown in two corollaries.

CH. Hirsch, B. Jahnel, P. Keeler, R.I.A. Patterson, Large deviations in relay-augmented wireless networks, Queueing Systems. Theory and Applications, 88 (2018), pp. 349--387 (published online on 28.10.2017).
Abstract
We analyze a model of relay-augmented cellular wireless networks. The network users, who move according to a general mobility model based on a Poisson point process of continuous trajectories in a bounded domain, try to communicate with a base station located at the origin. Messages can be sent either directly or indirectly by relaying over a second user. We show that in a scenario of an increasing number of users, the probability that an atypically high number of users experiences bad quality of service over a certain amount of time, decays at an exponential speed. This speed is characterized via a constrained entropy minimization problem. Further, we provide simulation results indicating that solutions of this problem are potentially non-unique due to symmetry breaking. Also two general sources for bad quality of service can be detected, which we refer to as isolation and screening.

CH. Hirsch, B. Jahnel, P. Keeler, R.I.A. Patterson, Traffic flow densities in large transport networks, Advances in Applied Probability, 49 (2017), pp. 1091--1115, DOI 10.1017/apr.2017.35 .
Abstract
We consider transport networks with nodes scattered at random in a large domain. At certain local rates, the nodes generate traffic flowing according to some navigation scheme in a given direction. In the thermodynamic limit of a growing domain, we present an asymptotic formula expressing the local traffic flow density at any given location in the domain in terms of three fundamental characteristics of the underlying network: the spatial intensity of the nodes together with their traffic generation rates, and of the links induced by the navigation. This formula holds for a general class of navigations satisfying a link-density and a sub-ballisticity condition. As a specific example, we verify these conditions for navigations arising from a directed spanning tree on a Poisson point process with inhomogeneous intensity function.

K.F. Lee, M. Dosta, A. Mc Guire, S. Mosbach, W. Wagner, S. Heinrich, M. Kraft, Development of a multi-compartment population balance model for high-shear wet granulation with discrete element method, Comput. Chem. Engng., 99 (2017), pp. 171--184.

O. Gün, A. Yilmaz, The stochastic encounter-mating model, Acta Applicandae Mathematicae. An International Survey Journal on Applying Mathematics and Mathematical Applications, 148 (2017), pp. 71--102.

B. Jahnel, Ch. Külske, The Widom--Rowlinson model under spin flip: Immediate loss and sharp recovery of quasilocality, The Annals of Applied Probability, 27 (2017), pp. 3845--3892, DOI 10.1214/17-AAP1298 .
Abstract
We consider the continuum Widom-Rowlinson model under independent spin-flip dynamics and investigate whether and when the time-evolved point process has an (almost) quasilocal specification (Gibbs-property of the time-evolved measure). Our study provides a first analysis of a Gibbs-non-Gibbs transition for point particles in Euclidean space. We find a picture of loss and recovery, in which even more regularity is lost faster than it is for time-evolved spin models on lattices. We show immediate loss of quasilocality in the percolation regime, with full measure of discontinuity points for any specification. For the color-asymmetric percolating model, there is a transition from this non-a.s. quasilocal regime back to an everywhere Gibbsian regime. At the sharp reentrance time t_G > 0 the model is a.s. quasilocal. For the colorsymmetric model there is no reentrance. On the constructive side, for all t > t_G , we provide everywhere quasilocal specifications for the time-evolved measures and give precise exponential estimates on the influence of boundary conditions.

W. König, Ch. Mukherjee, Mean-field interaction of Brownian occupation measures. I: Uniform tube property of the Coulomb functional, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 53 (2017), pp. 2214--2228, DOI 10.1214/16-AIHP788 .
Abstract
We study the transformed path measure arising from the self-interaction of a three-dimensional Brownian motion via an exponential tilt with the Coulomb energy of the occupation measures of the motion by time $t$. The logarithmic asymptotics of the partition function were identified in the 1980s by Donsker and Varadhan [DV83-P] in terms of a variational formula. Recently [MV14] a new technique for studying the path measure itself was introduced, which allows for proving that the normalized occupation measure asymptotically concentrates around the set of all maximizers of the formula. In the present paper, we show that likewise the Coulomb functional of the occupation measure concentrates around the set of corresponding Coulomb functionals of the maximizers in the uniform topology. This is a decisive step on the way to a rigorous proof of the convergence of the normalized occupation measures towards an explicit mixture of the maximizers, which will be carried out elsewhere. Our methods rely on deriving Hölder-continuity of the Coulomb functional of the occupation measure with exponentially small deviation probabilities and invoking the large-deviation theory developed in [MV14] to a certain shift-invariant functional of the occupation measures.

W. König, (Book review:) Firas Rassoul-Agha and Timo Seppäläinen: A Course on Large Deviations with an Introduction to Gibbs Measures, Jahresbericht der Deutschen Mathematiker-Vereinigung, 119 (2017), pp. 63--67.

A. Mielke, R.I.A. Patterson, M.A. Peletier, D.R.M. Renger, Non-equilibrium thermodynamical principles for chemical reactions with mass-action kinetics, SIAM Journal on Applied Mathematics, 77 (2017), pp. 1562--1585, DOI 10.1137/16M1102240 .
Abstract
We study stochastic interacting particle systems that model chemical reaction networks on the micro scale, converging to the macroscopic Reaction Rate Equation. One abstraction level higher, we study the ensemble of such particle systems, converging to the corresponding Liouville transport equation. For both systems, we calculate the corresponding large deviations and show that under the condition of detailed balance, the large deviations induce a non-linear relation between thermodynamic fluxes and free energy driving force.

R.I.A. Patterson, S. Simonella, W. Wagner, A kinetic equation for the distribution of interaction clusters in rarefied gases, Journal of Statistical Physics, 169 (2017), pp. 126--167.

A. González Casanova Soberón, N. Kurt, A. Wakolbinger, L. Yuan, An individual-based mathematical model for the Lenski experiment, and the deceleration of the relative fitness, Stochastic Processes and their Applications, 126 (2016), pp. 2211--2252.

CH. Hirsch, B. Jahnel, P. Keeler, R.I.A. Patterson, Large-deviation principles for connectable receivers in wireless networks, Advances in Applied Probability, 48 (2016), pp. 1061--1094.
Abstract
We study large-deviation principles for a model of wireless networks consisting of Poisson point processes of transmitters and receivers, respectively. To each transmitter we associate a family of connectable receivers whose signal-to-interference-and-noise ratio is larger than a certain connectivity threshold. First, we show a large-deviation principle for the empirical measure of connectable receivers associated with transmitters in large boxes. Second, making use of the observation that the receivers connectable to the origin form a Cox point process, we derive a large-deviation principle for the rescaled process of these receivers as the connection threshold tends to zero. Finally, we show how these results can be used to develop importance-sampling algorithms that substantially reduce the variance for the estimation of probabilities of certain rare events such as users being unable to connect.

CH. Hirsch, On the absence of percolation in a line-segment based lilypond model, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 52 (2016), pp. 127--145.

B. Jahnel, Ch. Külske, Attractor properties of non-reversible dynamics w.r.t. invariant Gibbs measures on the lattice, Markov Processes and Related Fields, 22 (2016), pp. 507--535.

P. Keeler, N. Ross, A. Xia, B. Błaszczyszyn, Stronger wireless signals appear more Poisson, IEEE Wireless Communications Letters, 5 (2016), pp. 572--575.
Abstract
Keeler, Ross and Xia [1] recently derived approximation and convergence results, which imply that the point process formed from the signal strengths received by an observer in a wireless network under a general statistical propagation model can be modelled by an inhomogeneous Poisson point process on the positive real line. The basic requirement for the results to apply is that there must be a large number of transmitters with different locations and random propagation effects. The aim of this note is to apply some of the main results of [1] in a less general but more easily applicable form to illustrate how the results can be applied in practice. New results are derived that show that it is the strongest signals, after being weakened by random propagation effects, that behave like a Poisson process, which supports recent experimental work.
[1] P. Keeler, N. Ross, and A. Xia:“When do wireless network signals appear Poisson?? ”

CH. Mukherjee, A. Shamov, O. Zeitouni, Weak and strong disorder for the stochastic heat equation and continuous directed polymers in $dgeq 3$, Electronic Communications in Probability, 21 (2016), pp. 1--12.

CH. Mukherjee, S.R.S. Varadhan, Brownian occupation measures, compactness and large deviations, The Annals of Probability, 44 (2016), pp. 3934--3964.
Abstract
In proving large deviation estimates, the lower bound for open sets and upper bound for compact sets are essentially local estimates. On the other hand, the upper bound for closed sets is global and compactness of space or an exponential tightness estimate is needed to establish it. In dealing with the occupation measure $L_t(A)=frac1tint_0^t1_A(W_s) d s$ of the $d$ dimensional Brownian motion, which is not positive recurrent, there is no possibility of exponential tightness. The space of probability distributions $mathcal M_1(R^d)$ can be compactified by replacing the usual topology of weak c onvergence by the vague toplogy, where the space is treated as the dual of continuous functions with compact support. This is essentially the one point compactification of $R^d$ by adding a point at $infty$ that results in the compactification of $mathcal M_1(R^d)$ by allowing some mass to escape to the point at $infty$. If one were to use only test functions that are continuous and vanish at $infty$ then the compactification results in the space of sub-probability distributions $mathcal M_le 1(R^d)$ by ignoring the mass at $infty$. The main drawback of this compactification is that it ignores the underlying translation invariance. More explicitly, we may be interested in the space of equivalence classes of orbits $widetildemathcal M_1=widetildemathcal M_1(R^d)$ under the action of the translation group $R^d$ on $mathcal M_1(R^d)$. There are problems for which it is natural to compactify this space of orbits. We will provide such a compactification, prove a large deviation principle there and give an application to a relevant problem.

H. Döring, G. Faraud, W. König, Connection times in large ad-hoc mobile networks, Bernoulli. Official Journal of the Bernoulli Society for Mathematical Statistics and Probability, 22 (2016), pp. 2143--2176.
Abstract
We study connectivity properties in a probabilistic model for a large mobile ad-hoc network. We consider a large number of participants of the system moving randomly, independently and identically distributed in a large domain, with a space-dependent population density of finite, positive order and with a fixed time horizon. Messages are instantly transmitted according to a relay principle, i.e., they are iteratedly forwarded from participant to participant over distances $leq 2R$, with $2R$ the communication radius, until they reach the recipient. In mathematical terms, this is a dynamic continuum percolation model. We consider the connection time of two sample participants, the amount of time over which these two are connected with each other. In the above thermodynamic limit, we find that the connectivity induced by the system can be described in terms of the counterplay of a local, random, and a global, deterministic mechanism, and we give a formula for the limiting behaviour. A prime example of the movement schemes that we consider is the well-known random waypoint model (RWP). Here we describe the decay rate, in the limit of large time horizons, of the probability that the portion of the connection time is less than the expectation.

V. Gayrard, O. Gün, Aging in the GREM-like trap model, Markov Processes and Related Fields, 22 (2016), pp. 165--202.
Abstract
The GREM-like trap model is a continuous time Markov jump process on the leaves of a finite volume L-level tree whose transition rates depend on a trapping landscape built on the vertices of the whole tree. We prove that the natural two-time correlation function of the dynamics ages in the infinite volume limit and identify the limiting function. Moreover, we take the limit L→ ∞ of the two-time correlation function of the infinite volume L-level tree. The aging behavior of the dynamics is characterized by a collection of clock processes, one for each level of the tree. We show that for any L, the joint law of the clock processes converges. Furthermore, any such limit can be expressed through Neveu's continuous state branching process. Hence, the latter contains all the information needed to describe aging in the GREM-like trap model both for finite and infinite levels.

M. Biskup, R. Fukushima, W. König, Eigenvalue fluctuations for lattice Anderson Hamiltonians, SIAM Journal on Mathematical Analysis, 48 (2016), pp. 2674--2700.
Abstract
We consider the random Schrödinger operator on a large box in the lattice with a large prefactor in front of the Laplacian part of the operator, which is proportional to the square of the diameter of the box. The random potential is assumed to be independent and bounded; its expectation function and variance function is given in terms of continuous bounded functions on the rescaled box. Our main result is a multivariate central limit theorem for all the simple eigenvalues of this operator, after centering and rescaling. The limiting covariances are expressed in terms of the limiting homogenized eigenvalue problem; more precisely, they are equal to the integral of the product of the squares of the eigenfunctions of that problem times the variance function.

M. Biskup, W. König, Eigenvalue order statistics for random Schrödinger operators with doubly-exponential tails, Communications in Mathematical Physics, 341 (2016), pp. 179--218.

J. Blath, A. González Casanova Soberón, N. Kurt, M. Wilke-Berenguer, A new coalescent for seed-bank models, The Annals of Applied Probability, 26 (2016), pp. 857--891.

E. Bouchet , Ch. Sabot, R. Soares Dos Santos, A quenched functional central limit theorem for random walks in random environments under (T)_gamma, Stochastic Processes and their Applications, 126 (2016), pp. 1206--1225.

A. Chiarini, A. Cipriani, R.S. Hazra, Extremes of some Gaussian random interfaces, Journal of Statistical Physics, 165 (2016), pp. 521--544.
Abstract
In this article we give a general criterion for some dependent Gaussian models to belong to maximal domain of attraction of Gumbel, following an application of the Stein-Chen method studied in citeAGG. We also show the convergence of the associated point process. As an application, we show the conditions are satisfied by some of the well-known supercritical Gaussian interface models, namely, membrane model, massive and massless discrete Gaussian free field, fractional Gaussian free field.

A. Chiarini, A. Cipriani, R.S. Hazra, Extremes of the supercritical Gaussian free field, ALEA. Latin American Journal of Probability and Mathematical Statistics, 13 (2016), pp. 711--724.
Abstract
We show that the rescaled maximum of the discrete Gaussian Free Field (DGFF) in dimension larger or equal to 3 is in the maximal domain of attraction of the Gumbel distribution. The result holds both for the infinite-volume field as well as the field with zero boundary conditions. We show that these results follow from an interesting application of the Stein-Chen method from Arratia et al. (1989).

T. Orenshtein, R. Soares Dos Santos, Zero-one law for directional transience of one-dimensional random walks in dynamic random environments, Electronic Communications in Probability, 21 (2016), pp. 15/1--15/11.
Abstract
We prove the trichotomy between transience to the right, transience to the left and recurrence of one-dimensional nearest-neighbour random walks in dynamic random environments under fairly general assumptions, namely: stationarity under space-time translations, ergodicity under spatial translations, and a mild ellipticity condition. In particular, the result applies to general uniformly elliptic models and also to a large class of non-uniformly elliptic cases that are i.i.d. in space and Markovian in time.

E.K.Y. Yapp, R.I.A. Patterson, J. Akroyd, S. Mosbach, E.M. Adkins, J.H. Miller, M. Kraft, Numerical simulation and parametric sensitivity study of optical band gap in a laminar co-flow ethylene diffusion flame, Combustion and Flame, 167 (2016), pp. 320--334.

A. Cipriani, A. Feidt, Rates of convergence for extremes of geometric random variables and marked point processes, Extremes. Statistical Theory and Applications in Science, Engineering and Economics, 19 (2016), pp. 105--138.
Abstract
We use the Stein-Chen method to study the extremal behaviour of univariate and bivariate geometric laws. We obtain a rate for the convergence, to the Gumbel distribution, of the law of the maximum of i.i.d. geometric random variables, and show that convergence is faster when approximating by a discretised Gumbel. We similarly find a rate of convergence for the law of maxima of bivariate Marshall-Olkin geometric random pairs when approximating by a discrete limit law. We introduce marked point processes of exceedances (MPPEs), both with univariate and bivariate Marshall-Olkin geometric variables as marks and determine bounds on the error of the approximation, in an appropriate probability metric, of the law of the MPPE by that of a Poisson process with same mean measure. We then approximate by another Poisson process with an easier-to-use mean measure and estimate the error of this additional approximation. This work contains and extends results contained in the second author's PhD thesis under the supervision of Andrew D. Barbour. The thesis is available at http://arxiv.org/abs/1310.2564.

R.I.A. Patterson, S. Simonella, W. Wagner, Kinetic theory of cluster dynamics, Physica D. Nonlinear Phenomena, 335 (2016), pp. 26--32.
Abstract
In a Newtonian system with localized interactions the whole set of particles is naturally decomposed into dynamical clusters, dened as nite groups of particles having an in uence on each other's trajectory during a given interval of time. For an ideal gas with shortrange intermolecular force, we provide a description of the cluster size distribution in terms of the reduced Boltzmann density. In the simplied context of Maxwell molecules, we show that a macroscopic fraction of the gas forms a giant component in nite kinetic time. The critical index of this phase transition is in agreement with previous numerical results on the elastic billiard.

W. Wagner, O. Muscato, A class of stochastic algorithms for the Wigner equation, SIAM Journal on Scientific Computing, 38 (2016), pp. A1483--A1507.
Abstract
A class of stochastic algorithms for the numerical treatment of the Wigner equation is introduced. The algorithms are derived using the theory of pure jump processes with a general state space. The class contains several new algorithms as well as some of the algorithms previously considered in the literature. The approximation error and the efficiency of the algorithms are analyzed. Numerical experiments are performed in a benchmark test case, where certain advantages of the new class of algorithms are demonstrated.

W. Wagner, A random cloud model for the Wigner equation, Kinetic and Related Models, 9 (2016), pp. 217--235.
Abstract
A probabilistic model for the Wigner equation is studied. The model is based on a particle system with the time evolution of a piecewise deterministic Markov process. Each particle is characterized by a real-valued weight, a position and a wave-vector. The particle position changes continuously, according to the velocity determined by the wave-vector. New particles are created randomly and added to the system. The main result is that appropriate functionals of the process satisfy a weak form of the Wigner equation.

CH. Hirsch, G.W. Delaney, V. Schmidt, Stationary Apollonian packings, Journal of Statistical Physics, 161 (2015), pp. 35--72.

CH. Hirsch, G. Gaiselmann, V. Schmidt, Asymptotic properties of collective-rearrangement algorithms, ESAIM. Probability and Statistics, 19 (2015), pp. 236--250.

CH. Hirsch, D. Neuhäuser, C. Gloaguen, V. Schmidt, Asymptotic properties of Euclidean shortest-path trees in random geometric graphs, Statistics & Probability Letters, 107 (2015), pp. 122--130.

CH. Hirsch, D. Neuhäuser, C. Gloaguen, V. Schmidt, First passage percolation on random geometric graphs and an application to shortest-path trees, Advances in Applied Probability, 47 (2015), pp. 328--354.

CH. Hirsch, A Harris--Kesten theorem for confetti percolation, Random Structures and Algorithms, 47 (2015), pp. 361--385.

B. Jahnel, Ch. Külske, A class of non-ergodic probabilistic cellular automata with unique invariant measure and quasi-periodic orbit, Stochastic Processes and their Applications, 125 (2015), pp. 2427--2450.

P. Keeler, P.G. Taylor, Discussion on ``On the Laplace transform of the aggregate discounted claims with Markovian arrivals'' by Jiandong Ren, Volume 12 (2), North American Actuarial Journal, 19 (2015), pp. 73--77.

S. Jansen, W. König, B. Metzger, Large deviations for cluster size distributions in a continuous classical many-body system, The Annals of Applied Probability, 25 (2015), pp. 930--973.
Abstract
An interesting problem in statistical physics is the condensation of classical particles in droplets or clusters when the pair-interaction is given by a stable Lennard-Jones-type potential. We study two aspects of this problem. We start by deriving a large deviations principle for the cluster size distribution for any inverse temperature $betain(0,infty)$ and particle density $rhoin(0,rho_rmcp)$ in the thermodynamic limit. Here $rho_rmcp >0$ is the close packing density. While in general the rate function is an abstract object, our second main result is the $Gamma$-convergence of the rate function towards an explicit limiting rate function in the low-temperature dilute limit $betatoinfty$, $rho downarrow 0$ such that $-beta^-1logrhoto nu$ for some $nuin(0,infty)$. The limiting rate function and its minimisers appeared in recent work, where the temperature and the particle density were coupled with the particle number. In the de-coupled limit considered here, we prove that just one cluster size is dominant, depending on the parameter $nu$. Under additional assumptions on the potential, the $Gamma$-convergence along curves can be strengthened to uniform bounds, valid in a low-temperature, low-density rectangle.

B. Blaszczyszyn, P. Keeler, Studying the SINR process of the typical user in Poisson networks by using its factorial moment measures, IEEE Transactions on Information Theory, 61 (2015), pp. 6774--6794.

B. Blaszczyszyn, M. Karray, P. Keeler, Wireless networks appear Poissonian due to strong shadowing, IEEE Transactions on Wireless Communications, 14 (2015), pp. 4379--4390.

A. Chiarini, A. Cipriani, R.S. Hazra, A note on the extremal process of the supercritical Gaussian free field, Electronic Communications in Probability, 20 (2015), pp. 74/1--74/10.
Abstract
We consider both the infinite-volume discrete Gaussian Free Field (DGFF) and the DGFF with zero boundary conditions outside a finite boxin dimension larger or equal to 3. We show that the associated extremal process converges to a Poisson point process. The result follows from an application of the Stein-Chen method from Arratia et al. (1989).

D. Neuhäuser, Ch. Hirsch, C. Gloaguen, V. Schmidt, Parametric modeling of sparse random trees using 3D copulas, Stochastic Models, 31 (2015), pp. 226--260.

K.F. Lee, S. Mosbach, M. Kraft, W. Wagner, A multi-compartment population balance model for high shear granulation, Comput. Chem. Engng., 75 (2015), pp. 1--13.
Abstract
This work extends the granulation model published by Braumann et al. (2007) to include multiple compartments in order to account for mixture heterogeneity encountered in powder mixing processes. A stochastic weighted algorithm is adapted to solve the granulation model which includes simultaneous coalescence and breakage. Then, a new numerical method to solve stochastic reactor networks is devised. The numerical behaviour of the adapted stochastic weighted algorithm is compared against the existing direct simulation algorithm. Lastly, the performance of the new compartmental model is then investigated by comparing the predicted particle size distribution against an experimentally measured size distribution. It is found that the adapted stochastic weighted algorithm exhibits superior performance compared to the direct simulation algorithm and the multi-compartment model produces results with better agreement with the experimental results compared to the original single-compartment model.

K.F. Lee, R.I.A. Patterson, W. Wagner, M. Kraft, Stochastic weighted particle methods for population balance equations with coagulation, fragmentation and spatial inhomogeneity, Journal of Computational Physics, 303 (2015), pp. 1--18.

A. Cipriani, D. Zeindler, The limit shape of random permutations with polynomially growing cycle weights, ALEA. Latin American Journal of Probability and Mathematical Statistics, 12 (2015), pp. 971--999.
Abstract
In this work we are considering the behavior of the limit shape of Young diagrams associated to random permutations on the set {1, ... n} under a particular class of multiplicative measures. Our method is based on generating functions and complex analysis (saddle point method). We show that fluctuations near a point behave like a normal random variable and that the joint fluctuations at different points of the limiting shape have an unexpected dependence structure. We will also compare our approach with the so-called randomization of the cycle counts of permutations and we will study the convergence of the limit shape to a continuous stochastic process.

O. Gün, W. König, O. Sekulović, Moment asymptotics for multitype branching random walks in random environment, Journal of Theoretical Probability, 28 (2015), pp. 1726--1742.
Abstract
We study a discrete time multitype branching random walk on a finite space with finite set of types. Particles follow a Markov chain on the spatial space whereas offspring distributions are given by a random field that is fixed throughout the evolution of the particles. Our main interest lies in the averaged (annealed) expectation of the population size, and its long-time asymptotics. We first derive, for fixed time, a formula for the expected population size with fixed offspring distributions, which is reminiscent of a Feynman-Kac formula. We choose Weibull-type distributions with parameter 1/ρ_ij for the upper tail of the mean number of j type particles produced by an i type particle. We derive the first two terms of the long-time asymptotics, which are written as two coupled variational formulas, and interpret them in terms of the typical behavior of the system.

W. König, T. Wolff, Large deviations for the local times of a random walk among random conductances in a growing box, Special issue for Pastur's 75th birthday, Markov Processes and Related Fields, 21 (2015), pp. 591--638.
Abstract
We derive an annealed large deviation principle (LDP) for the normalised and rescaled local times of a continuous-time random walk among random conductances (RWRC) in a time-dependent, growing box in Z^d. We work in the interesting case that the conductances are positive, but may assume arbitrarily small values. Thus, the underlying picture of the principle is a joint strategy of small conductance values and large holding times of the walk. The speed and the rate function of our principle are explicit in terms of the lower tails of the conductance distribution as well as the time-dependent size of the box.
An interesting phase transition occurs if the thickness parameter of the conductance tails exceeds a certain threshold: for thicker tails, the random walk spreads out over the entire growing box, for thinner tails it stays confined to some bounded region. In fact, in the first case, the rate function turns out to be equal to the p-th power of the p-norm of the gradient of the square root for some 2d/(d+2) < p < 2. This extends the Donsker-Varadhan-Gärtner rate function for the local times of Brownian motion (with deterministic environment) from p=2 to these values.
As corollaries of our LDP, we derive the logarithmic asymptotics of the non-exit probability of the RWRC from the growing box, and the Lifshitz tails of the generator of the RWRC, the randomised Laplace operator. To contrast with the annealed, not uniformly elliptic case, we also provide an LDP in the quenched setting for conductances that are bounded and bounded away from zero. The main tool here is a spectral homogenisation result, based on a quenched invariance principle for the RWRC.

W. Wagner, A class of probabilistic models for the Schrödinger equation, Monte Carlo Methods and Applications, 21 (2015), pp. 121--137.

R. Soares Dos Santos, Non-trivial linear bounds for a random walk driven by a simple symmetric exclusion process, Electronic Journal of Probability, 19 (2014), pp. 1--18.
Abstract
Linear bounds are obtained for the displacement of a random walk in a dynamic random environment given by a one-dimensional simple symmetric exclusion process in equilibrium. The proof uses an adaptation of multiscale renormalisation methods of Kesten and Sidoravicius [11].

R.I.A. Patterson, W. Wagner, Cell size error in stochastic particle methods for coagulation equations with advection, SIAM Journal on Numerical Analysis, 52 (2014), pp. 424--442.
Abstract
The paper studies the approximation error in stochastic particle methods for spatially inhomogeneous population balance equations. The model includes advection, coagulation and inception. Sufficient conditions for second order approximation with respect to the spatial discretization parameter (cell size) are provided. Examples are given, where only first order approximation is observed.

W. Wagner, A random cloud model for the Schrödinger equation, Kinetic and Related Models, 7 (2014), pp. 361--379.

L. Avena, R. caps">Dos Santos, F. Völlering, Transient random walk in symmetric exclusion: Limit theorems and an Einstein relation, ALEA. Latin American Journal of Probability and Mathematical Statistics, 10 (2013), pp. 693--709.

W.J. Menz, R.I.A. Patterson, W. Wagner, M. Kraft, Application of stochastic weighted algorithms to a multidimensional silica particle model, Journal of Computational Physics, 248 (2013), pp. 221--234.
Abstract
This paper presents a detailed study of the numerical behaviour of stochastic weighted algorithms (SWAs) using the transition regime coagulation kernel and a multidimensional silica particle model. The implementation in the SWAs of the transition regime coagulation kernel and associated majorant rates is described. The silica particle model of Shekar et al. [S. Shekar, A.J. Smith, W.J. Menz, M. Sander, M. Kraft, A multidimensional population balance model to describe the aerosol synthesis of silica nanoparticles, Journal of Aerosol Science 44 (2012) 83?98] was used in conjunction with this coagulation kernel to study the convergence properties of SWAs with a multidimensional particle model. High precision solutions were calculated with two SWAs and also with the established direct simulation algorithm. These solutions, which were generated using large number of computational particles, showed close agreement. It was thus demonstrated that SWAs can be successfully used with complex coagulation kernels and high dimensional particle models to simulate real-world systems.

O. Muscato, V. Di Stefano, W. Wagner, A variance-reduced electrothermal Monte Carlo method for semiconductor device simulation, Computers & Mathematics with Applications. An International Journal, 65 (2013), pp. 520--527.
Abstract
This paper is concerned with electron transport and heat generation in semiconductor devices. An improved version of the electrothermal Monte Carlo method is presented. This modification has better approximation properties due to reduced statistical fluctuations. The corresponding transport equations are provided and results of numerical experiments are presented.

W. König, Ch. Mukherjee, Large deviations for Brownian intersection measures, Communications on Pure and Applied Mathematics, 66 (2013), pp. 263--306.
Abstract
We consider $p$ independent Brownian motions in $R^d$. We assume that $pgeq 2$ and $p(d-2)<d$. Let $ell_t$ denote the intersection measure of the $p$ paths by time $t$, i.e., the random measure on $R^d$ that assigns to any measurable set $Asubset R^d$ the amount of intersection local time of the motions spent in $A$ by time $t$. Earlier results of Chen citeCh09 derived the logarithmic asymptotics of the upper tails of the total mass $ell_t(R^d)$ as $ttoinfty$. In this paper, we derive a large-deviation principle for the normalised intersection measure $t^-pell_t$ on the set of positive measures on some open bounded set $BsubsetR^d$ as $ttoinfty$ before exiting $B$. The rate function is explicit and gives some rigorous meaning, in this asymptotic regime, to the understanding that the intersection measure is the pointwise product of the densities of the normalised occupation times measures of the $p$ motions. Our proof makes the classical Donsker-Varadhan principle for the latter applicable to the intersection measure. A second version of our principle is proved for the motions observed until the individual exit times from $B$, conditional on a large total mass in some compact set $Usubset B$. This extends earlier studies on the intersection measure by König and Mörters citeKM01,KM05.

O. Gün, W. König, O. Sekulovic, Moment asymptotics for branching random walks in random environment, Electronic Journal of Probability, 18 (2013), pp. 1--18.

R.I.A. Patterson, Convergence of stochastic particle systems undergoing advection and coagulation, Stochastic Analysis and Applications, 31 (2013), pp. 800--829.
Abstract
The convergence of stochastic particle systems representing physical advection, inflow, outflow and coagulation is considered. The problem is studied on a bounded spatial domain such that there is a general upper bound on the residence time of a particle. The laws on the appropriate Skorohod path space of the empirical measures of the particle systems are shown to be relatively compact. The paths charged by the limits are characterised as solutions of a weak equation restricted to functions taking the value zero on the outflow boundary. The limit points of the empirical measures are shown to have densities with respect to Lebesgue measure when projected on to physical position space. In the case of a discrete particle type space a strong form of the Smoluchowski coagulation equation with a delocalised coagulation interaction and an inflow boundary condition is derived. As the spatial discretisation is refined in the limit equations, the delocalised coagulation term reduces to the standard local Smoluchowski interaction.

W. Wagner, Some properties of the kinetic equation for electron transport in semiconductors, Kinetic and Related Models, 6 (2013), pp. 955--967.
Abstract
The paper studies the kinetic equation for electron transport in semiconductors. New formulas for the heat generation rate are derived by analyzing the basic scattering mechanisms. In addition, properties of the steady state distribution are discussed and possible extensions of the deviational particle Monte Carlo method to the area of electron transport are proposed.

S. Jansen, W. König, Ideal mixture approximation of cluster size distributions at low density, Journal of Statistical Physics, 147 (2012), pp. 963--980.
Abstract
We consider an interacting particle system in continuous configuration space. The pair interaction has an attractive part. We show that, at low density, the system behaves approximately like an ideal mixture of clusters (droplets): we prove rigorous bounds (a) for the constrained free energy associated with a given cluster size distribution, considered as an order parameter, (b) for the free energy, obtained by minimising over the order parameter, and (c) for the minimising cluster size distributions. It is known that, under suitable assumptions, the ideal mixture has a transition from a gas phase to a condensed phase as the density is varied; our bounds hold both in the gas phase and in the coexistence region of the ideal mixture.

M. Becker, W. König, Self-intersection local times of random walks: Exponential moments in subcritical dimensions, Probability Theory and Related Fields, 154 (2012), pp. 585--605.
Abstract
Fix $p>1$, not necessarily integer, with $p(d-2)0$ that are bounded from above, possibly tending to zero. The speed is identified in terms of mixed powers of $t$ and $theta_t$, and the precise rate is characterized in terms of a variational formula, which is in close connection to the it Gagliardo-Nirenberg inequality. As a corollary, we obtain a large-deviation principle for $ ell_t _p/(t r_t)$ for deviation functions $r_t$ satisfying $t r_tggE[ ell_t _p]$. Informally, it turns out that the random walk homogeneously squeezes in a $t$-dependent box with diameter of order $ll t^1/d$ to produce the required amount of self-intersections. Our main tool is an upper bound for the joint density of the local times of the walk.

G. Benarous, O. Gün, Universality and extremal aging for the dynamics of spin glasses on subexponential time scales, Communications on Pure and Applied Mathematics, 65 (2012), pp. 77--127.

S. Shekar, W.J. Menz, A.J. Smith, M. Kraft, W. Wagner, On a multivariate population balance model to describe the structure and composition of silica nanoparticles, Comput. Chem. Engng., 43 (2012), pp. 130--147.

G. Faraud, Y. Hu, Z. Shi, Almost sure convergence for stochastically biased random walk on trees, Probability Theory and Related Fields, 154 (2012), pp. 621--660.

W. König, M. Salvi, T. Wolff, Large deviations for the local times of a random walk among random conductances, Electronic Communications in Probability, 17 (2012), pp. 1--11.
Abstract
We derive an annealed large deviation principle for the normalised local times of a continuous-time random walk among random conductances in a finite domain in $Z^d$ in the spirit of Donsker-Varadhan citeDV75. We work in the interesting case that the conductances may assume arbitrarily small values. Thus, the underlying picture of the principle is a joint strategy of small values of the conductances and large holding times of the walk. The speed and the rate function of our principle are explicit in terms of the lower tails of the conductance distribution. As an application, we identify the logarithmic asymptotics of the lower tails of the principal eigenvalue of the randomly perturbed negative Laplace operator in the domain.

R.I.A. Patterson, W. Wagner, A stochastic weighted particle method for coagulation-advection problems, SIAM Journal on Scientific Computing, 34 (2012), pp. B290--B311.
Abstract
A spatially resolved stochastic weighted particle method for inception--coagulation--advection problems is presented. Convergence to a deterministic limit is briefly studied. Numerical experiments are carried out for two problems with very different coagulation kernels. These tests show the method to be robust and confirm the convergence properties. The robustness of the weighted particle method is shown to contrast with two Direct Simulation Algorithms which develop instabilities.

S. Adams, A. Collevecchio, W. König, A variational formula for the free energy of an interacting many-particle system, The Annals of Probability, 39 (2011), pp. 683--728.
Abstract
We consider $N$ bosons in a box in $R^d$ with volume $N/rho$ under the influence of a mutually repellent pair potential. The particle density $rhoin(0,infty)$ is kept fixed. Our main result is the identification of the limiting free energy, $f(beta,rho)$, at positive temperature $1/beta$, in terms of an explicit variational formula, for any fixed $rho$ if $beta$ is sufficiently small, and for any fixed $beta$ if $rho$ is sufficiently small. The thermodynamic equilibrium is described by the symmetrised trace of $rm e^-beta Hcal_N$, where $Hcal_N$ denotes the corresponding Hamilton operator. The well-known Feynman-Kac formula reformulates this trace in terms of $N$ interacting Brownian bridges. Due to the symmetrisation, the bridges are organised in an ensemble of cycles of various lengths. The novelty of our approach is a description in terms of a marked Poisson point process whose marks are the cycles. This allows for an asymptotic analysis of the system via a large-deviations analysis of the stationary empirical field. The resulting variational formula ranges over random shift-invariant marked point fields and optimizes the sum of the interaction and the relative entropy with respect to the reference process. In our proof of the lower bound for the free energy, we drop all interaction involving lq infinitely longrq cycles, and their possible presence is signalled by a loss of mass of the lq finitely longrq cycles in the variational formula. In the proof of the upper bound, we only keep the mass on the lq finitely longrq cycles. We expect that the precise relationship between these two bounds lies at the heart of Bose-Einstein condensation and intend to analyse it further in future.

W. Kirsch, B. Metzger, P. Müller, Random block operators, Journal of Statistical Physics, 143 (2011), pp. 1035--1054.
Abstract
We study fundamental spectral properties of random block operators that are common in the physical modelling of mesoscopic disordered systems such as dirty superconductors. Our results include ergodic properties, the location of the spectrum, existence and regularity of the integrated density of states, as well as Lifshits tails. Special attention is paid to the peculiarities arising from the block structure such as the occurrence of a robust gap in the middle of the spectrum. Without randomness in the off-diagonal blocks the density of states typically exhibits an inverse square-root singularity at the edges of the gap. In the presence of randomness we establish a Wegner estimate that is valid at all energies. It implies that the singularities are smeared out by randomness, and the density of states is bounded. We also show Lifshits tails at these band edges. Technically, one has to cope with a non-monotone dependence on the random couplings.

G.A. Radtke, N.G. Hadjiconstantinou, W. Wagner, Low-noise Monte Carlo simulation of the variable hard sphere gas, Physics of Fluids, 23 (2011), pp. 030606/1--030606/12.

M. Sander, R.I.A. Patterson, A. Braumann, A. Raj, M. Kraft, Developing the PAH-PP soot particle model using process informatics and uncertainty propagation, Proceedings of the Combustion Institute, 33 (2011), pp. 675--683.
Abstract
n this work we present the new PAH-PP soot model and use a data collaboration approach to determine some of its parameters. The model describes the formation, growth and oxidation of soot in laminar premixed flames. Soot particles are modelled as aggregates containing primary particles, which are built from polycyclic aromatic hydrocarbons (PAHs), the main building blocks of a primary particle (PP). The connectivity of the primary particles is stored and used to determine the rounding of the soot particles due to surface growth and condensation processes. Two neighbouring primary particles are replaced by one if the coalescence level between the two primary particles reaches a threshold. The model contains, like most of the other models, free parameters that are unknown a priori. The experimental premixed flame data from Zhao et al. [B. Zhao, Z. Yang, Z. Li, M.V. Johnston, H. Wang, Proc. Combust. Inst. 30 (2) (2005) 1441?1448] have been used to estimate the smoothing factor of soot particles, the growth factor of PAHs within particles and the soot density using a low discrepancy series method with a subsequent response surface optimisation. The optimised particle size distributions show good agreement with the experimental ones. The importance of a standardised data mining system in order to optimise models is underlined.

O. Muscato, W. Wagner, V. Di Stefano, Properties of the steady state distribution of electrons in semiconductors, Kinetic and Related Models, 4 (2011), pp. 809--829.
Abstract
This paper studies a Boltzmann transport equation with several electron-phonon scattering mechanisms, which describes the charge transport in semiconductors. The electric field is coupled to the electron distribution function via Poisson's equation. Both the parabolic and the quasi-parabolic band approximations are considered. The steady state behaviour of the electron distribution function is investigated by a Monte Carlo algorithm. More precisely, several nonlinear functionals of the solution are calculated that quantify the deviation of the steady state from a Maxwellian distribution with respect to the wave-vector. On the one hand, the numerical results illustrate known theoretical statements about the steady state and indicate possible directions for future studies. On the other hand, the nonlinear functionals provide tools that can be used in the framework of Monte Carlo algorithms for detecting regions in which the steady state distribution has a relatively simple structure, thus providing a basis for domain decomposition methods.

G. Faraud, A central limit theorem for random walk in random environment on a marked Galton--Watson tree, Electronic Journal of Probability, 16 (2011), pp. 174--215.

G. Faraud, Estimates on the speedup and slowdown for a diffusion in a drifted Brownian potential, Journal of Theoretical Probability, 24 (2011), pp. 194-239.

W. König, P. Schmid, Brownian motion in a truncated Weyl chamber, Markov Processes and Related Fields, 17 (2011), pp. 499--522.
Abstract
We examine the non-exit probability of a multidimensional Brownian motion from a growing truncated Weyl chamber. Different regimes are identified according to the growth speed, ranging from polynomial decay over stretched-exponential to exponential decay. Furthermore we derive associated large deviation principles for the empirical measure of the properly rescaled and transformed Brownian motion as the dimension grows to infinity. Our main tool is an explicit eigenvalue expansion for the transition probabilities before exiting the truncated Weyl chamber.

R.I.A. Patterson, W. Wagner, M. Kraft, Stochastic weighted particle methods for population balance equations, Journal of Computational Physics, 230 (2011), pp. 7456--7472.
Abstract
A class of stochastic algorithms for the numerical treatment of population balance equations is introduced. The algorithms are based on systems of weighted particles, in which coagulation events are modelled by a weight transfer that keeps the number of computational particles constant. The weighting mechanisms are designed in such a way that physical processes changing individual particles (such as growth, or other surface reactions) can be conveniently treated by the algorithms. Numerical experiments are performed for complex laminar premixed flame systems. Two members of the class of stochastic weighted particle methods are compared to each other and to a direct simulation algorithm. One weighted algorithm is shown to be consistently better than the other with respect to the statistical noise generated. Finally, run times to achieve fixed error tolerances for a real flame system are measured and the better weighted algorithm is found to be up to three times faster than the direct simulation algorithm.

W. Wagner, Stochastic models in kinetic theory, Physics of Fluids, 23 (2011), pp. 030602/1--030602/14.
Abstract
The paper is concerned with some aspects of stochastic modelling in kinetic theory. First, an overview of the role of particle models with random interactions is given. These models are important both in the context of foundations of kinetic theory and for the design of numerical algorithms in various engineering applications. Then, the class of jump processes with a finite number of states is considered. Two types of such processes are studied, where particles change their states either independently of each other (mono-molecular processes), or via binary interactions (bi-molecular processes). The relationship of these processes with corresponding kinetic equations is discussed. Equations are derived both for the average relative numbers of particles in a given state and for the fluctuations of these numbers around their averages. The simplicity of the models makes several aspects of the theory more transparent.

M. Aizenman, S. Jansen, P. Jung, Symmetry breaking in quasi-1D Coulomb systems, Annales Henri Poincare. A Journal of Theoretical and Mathematical Physics, 11 (2010), pp. 1453--1485.
Abstract
Quasi one-dimensional systems are systems of particles in domains which are of infinite extent in one direction and of uniformly bounded size in all other directions, e.g. on a cylinder of infinite length. The main result proven here is that for such particle systems with Coulomb interactions and neutralizing background, the so-called “jellium”, at any temperature and at any finite-strip width there is translation symmetry breaking. This extends the previous result on Laughlin states in thin, two-dimens The structural argument which is used here bypasses the question of whether the translation symmetry breaking is manifest already at the level of the one particle density function. It is akin to that employed by Aizenman and Martin (1980) for a similar statement concerning symmetry breaking at all temperatures in strictly one-dimensional Coulomb systems. The extension is enabled through bounds which establish tightness of finite-volume charge fluctuations.

A. Braumann, M. Kraft, W. Wagner, Numerical study of a stochastic particle algorithm solving a multidimensional population balance model for high shear granulation, Journal of Computational Physics, 229 (2010), pp. 7672--7691.

A. Collevecchio, W. König, P. Mörters, N. Sidorova, Phase transitions for dilute particle systems with Lennard--Jones potential, Communications in Mathematical Physics, 299 (2010), pp. 603--630.

W. König, P. Schmid, Random walks conditioned to stay in Weyl chambers of type C and D, Electronic Communications in Probability, (2010), pp. 286--295.

W. Wagner, Random and deterministic fragmentation models, Monte Carlo Methods and Applications, 16 (2010), pp. 399--420.

A. Weiss, Escaping the Brownian stalkers, Electronic Journal of Probability, 14 (2009), pp. 139-160.
Abstract
We propose a simple model for the behaviour of long-time investors on stock markets, consisting of three particles, which represent the current price of the stock, and the opinion of the buyers, or sellers resp., about the right trading price. As time evolves both groups of traders update their opinions with respect to the current price. The update speed is controled by a parameter $gamma$, the price process is described by a geometric Brownian motion. The stability of the market is governed by the difference of the buyers' opinion and the sellers' opinion. We prove that the distance

M. Becker, W. König, Moments and distribution of the local times of a transient random walk on $Bbb Zsp d$, Journal of Theoretical Probability, 22 (2009), pp. 365--374.

G. Grüninger, W. König, Potential confinement property in the parabolic Anderson model, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 45 (2009), pp. 840--863.

W. König, H. Lacoin, P. Mörters, N. Sidorova, A two cities theorem for the parabolic Anderson model, The Annals of Probability, 37 (2009), pp. 347--392.

R. Siegmund-Schultze, W. Wagner, Induced gelation in a two-site spatial coagulation model, The Annals of Applied Probability, 16 (2006), pp. 370-402.

W. Wagner, Post-gelation behavior of a spatial coagulation model, Electronic Journal of Probability, 11 (2006), pp. 893-933.

W. Wagner, Explosion phenomena in stochastic coagulation-fragmentation models, The Annals of Applied Probability, 15 (2005), pp. 2081-2112.

A. Eibeck, W. Wagner, Stochastic interacting particle systems and nonlinear kinetic equations, The Annals of Applied Probability, 13 (2003), pp. 845-889.

A. Hinsen, B. Jahnel, E. Cali, J.-P. Wary, Malware propagation in urban D2D networks, IEEE 18th International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt), Institut of Electrical and Electronics Engineer (IEEE), 2020, pp. 1--6.
Abstract
We introduce and analyze models for the propagation of malware in pure D2D networks given via stationary Cox--Gilbert graphs. Here, the devices form a Poisson point process with random intensity measure λ, Λ where Λ is stationary and given, for example, by the edge-length measure of a realization of a Poisson--Voronoi tessellation that represents an urban street system. We assume that, at initial time, a typical device at the center of the network carries a malware and starts to infect neighboring devices after random waiting times. Here we focus on Markovian models, where the waiting times are exponential random variables, and non-Markovian models, where the waiting times feature strictly positive minimal and finite maximal waiting times. We present numerical results for the speed of propagation depending on the system parameters. In a second step, we introduce and analyze a counter measure for the malware propagation given by special devices called white knights, which have the ability, once attacked, to eliminate the malware from infected devices and turn them into white knights. Based on simulations, we isolate parameter regimes in which the malware survives or is eliminated, both in the Markovian and non-Markovian setting.

CH. Hirsch, B. Jahnel, A. Hinsen, E. Cali, The typical cell in anisotropic tessellations, IEEE 17th International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOPT), Avignon, France, June 3 - 7, 2019, Institut of Electrical and Electronics Engineer (IEEE), 2019, pp. 1--6, DOI 10.23919/WiOPT47501.2019.9144122 .
Abstract
The typical cell is a key concept for stochastic-geometry based modeling in communication networks, as it provides a rigorous framework for describing properties of a serving zone associated with a component selected at random in a large network. We consider a setting where network components are located on a large street network. While earlier investigations were restricted to street systems without preferred directions, in this paper we derive the distribution of the typical cell in Manhattan-type systems characterized by a pattern of horizontal and vertical streets. We explain how the mathematical description can be turned into a simulation algorithm and provide numerical results uncovering novel effects when compared to classical isotropic networks.

B. Jahnel, W. König, Probabilistic methods for spatial multihop communication systems, in: Topics in Applied Analysis and Optimisation, M. Hintermüller, J.F. Rodrigues, eds., CIM Series in Mathematical Sciences, Springer Nature Switzerland AG, Cham, 2019, pp. 239--268.

E. Cali, N.N. Gafur, Ch. Hirsch, B. Jahnel, T. En-Najjari, R.I.A. Patterson, Percolation for D2D networks on street systems, in: 2018 16th International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt), IEEE Xplore digital library, 2018, pp. 17789106/1--17789106/6, DOI 10.23919/WIOPT.2018.8362866 .
Abstract
We study fundamental characteristics for the connectivity of multi-hop D2D networks. Devices are randomly distributed on street systems and are able to communicate with each other whenever their separation is smaller than some connectivity threshold. We model the street systems as Poisson-Voronoi or Poisson-Delaunay tessellations with varying street lengths. We interpret the existence of adequate D2D connectivity as percolation of the underlying random graph. We derive and compare approximations for the critical device-intensity for percolation, the percolation probability and the graph distance. Our results show that for urban areas, the Poisson Boolean Model gives a very good approximation, while for rural areas, the percolation probability stays far from 1 even far above the percolation threshold.

P. Keeler, B. Jahnel, O. Maye, D. Aschenbach, M. Brzozowski, Disruptive events in high-density cellular networks, in: 2018 16th International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt), IEEE Xplore digital library, 2018, pp. 17789136/1--17789136/8, DOI 10.23919/WIOPT.2018.8362867 .
Abstract
Stochastic geometry models are used to study wireless networks, particularly cellular phone networks, but most of the research focuses on the typical user, often ignoring atypical events, which can be highly disruptive and of interest to network operators. We examine atypical events when a unexpected large proportion of users are disconnected or connected by proposing a hybrid approach based on ray launching simulation and point process theory. This work is motivated by recent results [12] using large deviations theory applied to the signal-to-interference ratio. This theory provides a tool for the stochastic analysis of atypical but disruptive events, particularly when the density of transmitters is high. For a section of a European city, we introduce a new stochastic model of a single network cell that uses ray launching data generated with the open source RaLaNS package, giving deterministic path loss values. We collect statistics on the fraction of (dis)connected users in the uplink, and observe that the probability of an unexpected large proportion of disconnected users decreases exponentially when the transmitter density increases. This observation implies that denser networks become more stable in the sense that the probability of the fraction of (dis)connected users deviating from its mean, is exponentially small. We also empirically obtain and illustrate the density of users for network configurations in the disruptive event, which highlights the fact that such bottleneck behaviour not only stems from too many users at the cell boundary, but also from the near-far effect of many users in the immediate vicinity of the base station. We discuss the implications of these findings and outline possible future research directions.

J. Blath, E. Bjarki, A. González Casanova Soberón, N. Kurt, Genealogy of a Wright--Fisher model with strong seedbank component, in: XI Symposium of Probability and Stochastic Processes, R.H. Mena, J.C. Pardo, V. Rivero, G. Uribe Bravo, eds., 69 of Birkhäuser Progress in Probability, Springer International Publishing, Switzerland, 2015, pp. 81--100.

F. Castell, O. Gün, G. Maillard, Parabolic Anderson model with finite number of moving catalysts, in: Probability in Complex Physical Systems, in Honour of Erwin Bolthausen and Jürgen Gärtner, J.-D. Deuschel, B. Gentz, W. König, M. VON Renesse, M. Scheutzow, U. Schmock, eds., 11 of Springer Proceedings in Mathematics, Springer, Berlin Heidelberg, 2012, pp. 91--116.

A. Schnitzler, T. Wolff, Precise asymptotics for the parabolic Anderson model with a moving catalyst or trap, in: Probability in Complex Physical Systems, in Honour of Erwin Bolthausen and Jürgen Gärtner, J.-D. Deuschel, B. Gentz, W. König, M. VON Renesse, M. Scheutzow, eds., 11 of Springer Proceedings in Mathematics, Springer, Berlin Heidelberg, 2012, pp. 69--88.

W. König, S. Schmidt, The parabolic Anderson model with acceleration and deceleration, in: Probability in Complex Physical Systems, in Honour of Erwin Bolthausen and Jürgen Gärtner, J.-D. Deuschel, B. Gentz, W. König, M. VON Renesse, M. Scheutzow, U. Schmock, eds., 11 of Springer Proceedings in Mathematics, Springer, Berlin Heidelberg, 2012, pp. 225--245.

G.A. Radtke, N.G. Hadjiconstantinou, W. Wagner, Low variance particle simulations of the Boltzmann transport equation for the variable hard sphere collision model, in: 27th International Symposium on Rarefied Gas Dynamics, 2010, Pacific Grove, California, July 10-15, 2010, Part One, D.A. Levin, I.J. Wysong, A.L. Garcia, eds., 1333 of AIP Conference Proceedings, AIP Publishing Center, New York, 2011, pp. 307--312.

W. König, Upper tails of self-intersection local times of random walks: Survey of proof techniques, in: Excess Self-Intersections & Related Topics, 2 of Actes des Rencontres du CIRM, Centre International de Rencontres Mathématiques, Marseille, 2010, pp. 15--24.

W. König, Branching random walks in random environment: A survey, Preprint no. 2779, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2779 .
Abstract, PDF (253 kByte)
We consider branching particle processes on discrete structures like the hypercube in a random fitness landscape (i.e., random branching/killing rates). The main question is about the location where the main part of the population sits at a late time, if the state space is large. For answering this, we take the expectation with respect to the migration (mutation) and the branching/killing (selection) mechanisms, for fixed rates. This is intimately connected with the parabolic Anderson model, the heat equation with random potential, a model that is of interest in mathematical physics because of the observed prominent effect of intermittency (local concentration of the mass of the solution in small islands). We present several advances in the investigation of this effect, also related to questions inspired from biology.

B. Jahnel, A. Tóbiás, Absence of percolation in graphs based on stationary point processes with degrees bounded by two, Preprint no. 2774, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2774 .
Abstract, PDF (548 kByte)
We consider undirected graphs that arise as deterministic functions of stationary point processes such that each point has degree bounded by two. For a large class of point processes and edge-drawing rules, we show that the arising graph has no infinite connected component, almost surely. In particular, this extends our previous result for SINR graphs based on stabilizing Cox point processes and verifies the conjecture of Balister and Bollobás that the bidirectional $k$-nearest neighbor graph of a two-dimensional homogeneous Poisson point process does not percolate for k=2.

N. Perkowski, W. van Zuijlen, Quantitative heat kernel estimates for diffusions with distributional drift, Preprint no. 2768, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2768 .
Abstract, PDF (308 kByte)
We consider the stochastic differential equation on ℝ ^d given by d X _t = b(t,X_t ) d t + d B_t, where B is a Brownian motion and b is considered to be a distribution of regularity > - 1/2. We show that the martingale solution of the SDE has a transition kernel Γ_t and prove upper and lower heat kernel bounds for Γ_t with explicit dependence on t and the norm of b.

M.A. Peletier, D.R.M. Renger, Fast reaction limits via $Gamma$-convergence of the flux rate functional, Preprint no. 2766, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2766 .
Abstract, PDF (492 kByte)
We study the convergence of a sequence of evolution equations for measures supported on the nodes of a graph. The evolution equations themselves can be interpreted as the forward Kolmogorov equations of Markov jump processes, or equivalently as the equations for the concentrations in a network of linear reactions. The jump rates or reaction rates are divided in two classes; `slow' rates are constant, and `fast' rates are scaled as 1/∈, and we prove the convergence in the fast-reaction limit ∈ → 0. We establish a Γ-convergence result for the rate functional in terms of both the concentration at each node and the flux over each edge (the level-2.5 rate function). The limiting system is again described by a functional, and characterizes both fast and slow fluxes in the system. This method of proof has three advantages. First, no condition of detailed balance is required. Secondly, the formulation in terms of concentration and flux leads to a short and simple proof of the Γ-convergence; the price to pay is a more involved compactness proof. Finally, the method of proof deals with approximate solutions, for which the functional is not zero but small, without any changes.

W. König, N. Perkowski, W. van Zuijlen, Longtime asymptotics of the two-dimensional parabolic Anderson model with white-noise potential, Preprint no. 2765, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2765 .
Abstract, PDF (471 kByte)
We consider the parabolic Anderson model (PAM) in ℝ ² with a Gaussian (space) white-noise potential. We prove that the almost-sure large-time asymptotic behaviour of the total mass at time t is given asymptotically by Χ t log t, with the deterministic constant Χ identified in terms of a variational formula. In earlier work of one of the authors this constant was used to describe the asymptotic behaviour principal Dirichlet of the eigenvalue the Anderson operator on the t by t box around zero asymptotically by Χ log t.

J.-D. Deuschel, T. Orenshtein, G.R. Moreno Flores, Aging for the stationary Kardar--Parisi--Zhang equation and related models, Preprint no. 2763, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2763 .
Abstract, PDF (368 kByte)
We study the aging property for stationary models in the KPZ universality class. In particular, we show aging for the stationary KPZ fixed point, the Cole-Hopf solution to the stationary KPZ equation, the height function of the stationary TASEP, last-passage percolation with boundary conditions and stationary directed polymers in the intermediate disorder regime. All of these models are shown to display a universal aging behavior characterized by the rate of decay of their correlations. As a comparison, we show aging for models in the Edwards-Wilkinson universality class where a different decay exponent is obtained. A key ingredient to our proofs is a characteristic of space-time stationarity - covariance-to-variance reduction - which allows to deduce the asymptotic behavior of the correlations of two space-time points by the one of the variances at one point. We formulate several open problems.

D. Gabrielli, D.R.M. Renger, Dynamical phase transitions for flows on finite graphs, Preprint no. 2746, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2746 .
Abstract, PDF (304 kByte)
We study the time-averaged flow in a model of particles that randomly hop on a finite directed graph. In the limit as the number of particles and the time window go to infinity but the graph remains finite, the large-deviation rate functional of the average flow is given by a variational formulation involving paths of the density and flow. We give sufficient conditions under which the large deviations of a given time averaged flow is determined by paths that are constant in time. We then consider a class of models on a discrete ring for which it is possible to show that a better strategy is obtained producing a time-dependent path. This phenomenon, called a dynamical phase transition, is known to occur for some particle systems in the hydrodynamic scaling limit, which is thus extended to the setting of a finite graph.

B. Lees, L. Taggi, Exponential decay of transverse correlations for spin systems with continuous symmetry and non-zero external field, Preprint no. 2730, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2730 .
Abstract, PDF (381 kByte)
We prove exponential decay of transverse correlations in the Spin O(N) model for arbitrary (non-zero) values of the external magnetic field and arbitrary spin dimension N > 1. Our result is new when N > 3, in which case no Lee-Yang theorem is available, it is an alternative to Lee-Yang when N = 2, 3, and also holds for a wide class of multi-component spin systems with continuous symmetry. The key ingredients are a representation of the model as a system of coloured random paths, a `colour-switch' lemma, and a sampling procedure which allows us to bound from above the `typical' length of the open paths.

L. Taggi, Essential enhancements in Abelian networks: Continuity and uniform strict monotonicity, Preprint no. 2722, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2722 .
Abstract, PDF (311 kByte)
We prove that in wide generality the critical curve of the activated random walk model is a continuous function of the deactivation rate, and we provide a bound on its slope which is uniform with respect to the choice of the graph. Moreover, we derive strict monotonicity properties for the probability of a wide class of `increasing' events, extending previous results of Rolla and Sidoravicius (2012). Our proof method is of independent interest and can be viewed as a reformulation of the `essential enhancements' technique -- which was introduced for percolation -- in the framework of Abelian networks.

B. Lees, L. Taggi, Site-monotonicity properties for reflection positive measures with applications to quantum spin systems, Preprint no. 2713, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2713 .
Abstract, PDF (298 kByte)
We consider a general statistical mechanics model on a product of local spaces and prove that, if the corresponding measure is reflection positive, then several site-monotonicity properties for the two-point function hold. As an application of such a general theorem, we derive site-monotonicity properties for the spin-spin correlation of the quantum Heisenberg antiferromagnet and XY model, we prove that such spin-spin correlations are point-wise uniformly positive on vertices with all odd coordinates -- improving previous positivity results which hold for the Cesàro sum -- and we derive site-monotonicity properties for the probability that a loop connects two vertices in various random loop models, including the loop representation of the spin O(N) model, the double-dimer model, the loop O(N) model, lattice permutations, thus extending the previous results of Lees and Taggi (2019).

B. Jahnel, A. Tóbiás, E. Cali, Phase transitions for the Boolean model of continuum percolation for Cox point processes, Preprint no. 2704, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2704 .
Abstract, PDF (389 kByte)
We consider the Boolean model with random radii based on Cox point processes. Under a condition of stabilization for the random environment, we establish existence and non-existence of subcritical regimes for the size of the cluster at the origin in terms of volume, diameter and number of points. Further, we prove uniqueness of the infinite cluster for sufficiently connected environments.

J.-D. Deuschel, T. Orenshtein, N. Perkowski, Additive functionals as rough paths, Preprint no. 2685, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2685 .
Abstract, PDF (335 kByte)
We consider additive functionals of stationary Markov processes and show that under Kipnis--Varadhan type conditions they converge in rough path topology to a Stratonovich Brownian motion, with a correction to the Lévy area that can be described in terms of the asymmetry (non-reversibility) of the underlying Markov process. We apply this abstract result to three model problems: First we study random walks with random conductances under the annealed law. If we consider the Itô rough path, then we see a correction to the iterated integrals even though the underlying Markov process is reversible. If we consider the Stratonovich rough path, then there is no correction. The second example is a non-reversible Ornstein-Uhlenbeck process, while the last example is a diffusion in a periodic environment. As a technical step we prove an estimate for the p-variation of stochastic integrals with respect to martingales that can be viewed as an extension of the rough path Burkholder-Davis-Gundy inequality for local martingale rough paths of [FV08], [CF19] and [FZ18] to the case where only the integrator is a local martingale.

F. DEN Hollander, W. König, R. Soares Dos Santos, The parabolic Anderson model on a Galton--Watson tree, Preprint no. 2675, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2675 .
Abstract, PDF (428 kByte)
We study the long-time asymptotics of the total mass of the solution to the parabolic Anderson model ( PAM) on a supercritical Galton-Watson random tree with bounded degrees. We identify the second-order contribution to this asymptotics in terms of a variational formula that gives information about the local structure of the region where the solution is concentrated. The analysis behind this formula suggests that, under mild conditions on the model parameters, concentration takes place on a tree with minimal degree. Our approach can be applied to finite locally tree-like random graphs, in a coupled limit where both time and graph size tend to infinity. As an example, we consider the configuration model or, more precisely, the uniform simple random graph with a prescribed degree sequence.

A. Hinsen, B. Jahnel, E. Cali, J.-P. Wary, Malware propagation in urban D2D networks, Preprint no. 2674, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2674 .
Abstract, PDF (3133 kByte)
We introduce and analyze models for the propagation of malware in pure D2D networks given via stationary Cox--Gilbert graphs. Here, the devices form a Poisson point process with random intensity measure λ, Λ where Λ is stationary and given, for example, by the edge-length measure of a realization of a Poisson--Voronoi tessellation that represents an urban street system. We assume that, at initial time, a typical device at the center of the network carries a malware and starts to infect neighboring devices after random waiting times. Here we focus on Markovian models, where the waiting times are exponential random variables, and non-Markovian models, where the waiting times feature strictly positive minimal and finite maximal waiting times. We present numerical results for the speed of propagation depending on the system parameters. In a second step, we introduce and analyze a counter measure for the malware propagation given by special devices called white knights, which have the ability, once attacked, to eliminate the malware from infected devices and turn them into white knights. Based on simulations, we isolate parameter regimes in which the malware survives or is eliminated, both in the Markovian and non-Markovian setting.

B. Jahnel, A. Tóbiás, SINR percolation for Cox point processes with random powers, Preprint no. 2659, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2659 .
Abstract, PDF (356 kByte)
Signal-to-interference plus noise ratio (SINR) percolation is an infinite-range dependent variant of continuum percolation modeling connections in a telecommunication network. Unlike in earlier works, in the present paper the transmitted signal powers of the devices of the network are assumed random, i.i.d. and possibly unbounded. Additionally, we assume that the devices form a stationary Cox point process, i.e., a Poisson point process with stationary random intensity measure, in two or higher dimensions. We present the following main results. First, under suitable moment conditions on the signal powers and the intensity measure, there is percolation in the SINR graph given that the device density is high and interferences are sufficiently reduced, but not vanishing. Second, if the interference cancellation factor γ and the SINR threshold τ satisfy γ ≥ 1/(2τ), then there is no percolation for any intensity parameter. Third, in the case of a Poisson point process with constant powers, for any intensity parameter that is supercritical for the underlying Gilbert graph, the SINR graph also percolates with some small but positive interference cancellation factor.

L. Taggi, Uniformly positive correlations in the dimer model and phase transition in lattice permutations in $mathbbZ^d, d > 2$, via reflection positivity, Preprint no. 2647, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2647 .
Abstract, PDF (580 kByte)
Our first main result is that correlations between monomers in the dimer model in ℤ^d do not decay to zero when d > 2. This is the first rigorous result about correlations in the dimer model in dimensions greater than two and shows that the model behaves drastically differently than in two dimensions, in which case it is integrable and correlations are known to decay to zero polynomially. Such a result is implied by our more general, second main result, which states the occurrence of a phase transition in the model of lattice permutations, which is related to the quantum Bose gas. More precisely, we consider a self-avoiding walk interacting with lattice permutations and we prove that, in the regime of fully-packed loops, such a walk is `long' and the distance between its end-points grows linearly with the diameter of the box. These results follow from the derivation of a version of the infrared bound from a new general probabilistic settings, with coloured loops and walks interacting at sites and walks entering into the system from some `virtual' vertices.

A. Hinsen, B. Jahnel, E. Cali, J.-P. Wary, Phase transitions for chase-escape models on Gilbert graphs, Preprint no. 2642, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2642 .
Abstract, PDF (219 kByte)
We present results on phase transitions of local and global survival in a two-species model on Gilbert graphs. At initial time there is an infection at the origin that propagates on the Gilbert graph according to a continuous-time nearest-neighbor interacting particle system. The Gilbert graph consists of susceptible nodes and nodes of a second type, which we call white knights. The infection can spread on susceptible nodes without restriction. If the infection reaches a white knight, this white knight starts to spread on the set of infected nodes according to the same mechanism, with a potentially different rate, giving rise to a competition of chase and escape. We show well-definedness of the model, isolate regimes of global survival and extinction of the infection and present estimates on local survival. The proofs rest on comparisons to the process on trees, percolation arguments and finite-degree approximations of the underlying random graphs.

D. Heydecker, R.I.A. Patterson, Bilinear coagulation equations, Preprint no. 2637, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2637 .
Abstract, PDF (453 kByte)
We consider coagulation equations of Smoluchowski or Flory type where the total merge rate has a bilinear form π(y) · Aπ (x) for a vector of conserved quantities π, generalising the multiplicative kernel. For these kernels, a gelation transition occurs at a finite time t_g ∈ (0,∞), which can be given exactly in terms of an eigenvalue problem in finite dimensions. We prove a hydrodynamic limit for a stochastic coagulant, including a corresponding phase transition for the largest particle, and exploit a coupling to random graphs to extend analysis of the limiting process beyond the gelation time.

K. Chouk, W. van Zuijlen, Asymptotics of the eigenvalues of the Anderson Hamiltonian with white noise potential in two dimensions, Preprint no. 2606, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2606 .
Abstract, PDF (588 kByte)
In this paper we consider the Anderson Hamiltonian with white noise potential on the box [0,L]² with Dirichlet boundary conditions. We show that all the eigenvalues divided by log L converge as L → ∞ almost surely to the same deterministic constant, which is given by a variational formula.

S. Jansen, W. König, B. Schmidt, F. Theil, Surface energy and boundary layers for a chain of atoms at low temperature, Preprint no. 2589, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2589 .
Abstract, PDF (529 kByte)
We analyze the surface energy and boundary layers for a chain of atoms at low temperature for an interaction potential of Lennard-Jones type. The pressure (stress) is assumed small but positive and bounded away from zero, while the temperature goes to zero. Our main results are: (1) As  the temperature goes to zero and at fixed positive pressure, the Gibbs measures for infinite chains and semi-infinite chains satisfy path large deviations principles. The rate functions are bulk and surface energy functionals. The minimizer of the surface functional corresponds to zero temperature boundary layers. (2) The surface correction to the Gibbs free energy converges to the zero temperature surface energy, characterized with the help of the minimum of the surface energy functional. (3) The bulk Gibbs measure and Gibbs free energy can be approximated by their Gaussian counterparts. (4) Bounds on the decay of correlations are provided, some of them uniform in the inverse temperature.

L. Andreis, W. König, R.I.A. Patterson, A large-deviations approach to gelation, Preprint no. 2568, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2568 .
Abstract, PDF (338 kByte)
A large-deviations principle (LDP) is derived for the state, at fixed time, of the multiplicative coalescent in the large particle number limit. The rate function is explicit and describes each of the three parts of the state: microscopic, mesoscopic and macroscopic. In particular, it clearly captures the well known gelation phase transition given by the formation of a particle containing a positive fraction of the system mass at time t=1. Via a standard map of the multiplicative coalescent onto a time-dependent version of the Erdős-Rényi random graph, our results can also be rephrased as an LDP for the component sizes in that graph. Our proofs rely on estimates and asymptotics for the probability that smaller Erdős-Rényi graphs are connected.

CH. Hirsch, B. Jahnel, A. Hinsen, E. Cali, The typical cell in anisotropic tessellations, Preprint no. 2557, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2557 .
Abstract, PDF (311 kByte)
The typical cell is a key concept for stochastic-geometry based modeling in communication networks, as it provides a rigorous framework for describing properties of a serving zone associated with a component selected at random in a large network. We consider a setting where network components are located on a large street network. While earlier investigations were restricted to street systems without preferred directions, in this paper we derive the distribution of the typical cell in Manhattan-type systems characterized by a pattern of horizontal and vertical streets. We explain how the mathematical description can be turned into a simulation algorithm and provide numerical results uncovering novel effects when compared to classical isotropic networks.

F. Flegel, Eigenvector localization in the heavy-tailed random conductance model, Preprint no. 2472, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2472 .
Abstract, PDF (292 kByte)
We generalize our former localization result about the principal Dirichlet eigenvector of the i.i.d. heavy-tailed random conductance Laplacian to the first k eigenvectors. We overcome the complication that the higher eigenvectors have fluctuating signs by invoking the Bauer-Fike theorem to show that the kth eigenvector is close to the principal eigenvector of an auxiliary spectral problem.

A. Quitmann, Probabilistic treatment of Bose--Einstein Condensation, Summer School 2020: Jahrestreffen des IRTG 2544, September 14 - 17, 2020, Stochastic Analysis in Interaction. Berlin--Oxford IRTG 2544, Döllnsee.

A. Hinsen, Malware propagation in urban D2D networks., The 14th Workshop on Spatial Stochastic Models for Wireless Networks (SPASWIN), June 19, 2020, online event, Greece, June 19, 2020.

T. Orenshtein, Aging for the stationary KPZ equation, The 3rd Haifa Probability School. Workshop on Random Geometry and Stochastik Analysis, February 24 - 28, 2020, Technion Israel Institute of Technology, Haifa, Israel, February 25, 2020.

T. Orenshtein, Functional Inequalities on sub-Riemannian manifolds via QCD, Geometric Functional Analysis and Probability Seminar, The Weizmann Institute of Science, Faculty of Mathematics and Computer Science, Rechovot, Israel, March 5, 2020.

M. Renger, Dynamical Phase Transitions on Finite Graphs (online), DMV Jahrestagung 2020, September 14 - October 17, 2020, Technische Universität Chemnitz, Chemnitz, September 15, 2020.

W. van Zuijlen, Large time behaviour of the parabolic Anderson model, Probability Seminar, Universidade Federal da Bahia, Instituto de Matematica Doutorado em Matematica, Salvador, Brazil, October 21, 2020.

W. van Zuijlen, Spectral asymptotics of the Anderson Hamiltonian, Forschungsseminar ''Functional Analysis``, Karlsruher Institut für Technologie, Fakultät für Mathematik, Institut für Analysis, January 21, 2020.

L. Andreis, A large deviations approach to sparse random graphs, Bernoulli--IMS One World Symposium 2020, August 24 - 28, 2020, online Vortrag, August 25, 2020.

L. Andreis, A large-deviations approach to the phase transitionin inhomogeneous random graphs: part II, Spring School: Complex Networks, March 2 - 6, 2020, Technische Universität Darmstadt, Fachbereich Mathematik, March 2, 2020.

L. Andreis, Phase transitions in inhomogeneous random graphs and coagulation processes, The 3rd Haifa Probability School. Workshop on Random Geometry and Stochastik Analysis, February 24 - 28, 2020, Technion Israel Institute of Technology, Haifa, Israel, February 25, 2020.

B. Jahnel, Phase transitions for the Boolean model for Cox point processes, Bernoulli-IMS One World Symposium 2020, August 24 - 28, 2020, A virtual one week symposium on Probability and Mathematical Statistics, August 27, 2020.

W. König, Probabilistic treatment of Bose--Einstein Condensation, Summer School 2020: Jahrestreffen des IRTG 2544, September 14 - 17, 2020, Stochastic Analysis in Interaction. Berlin--Oxford IRTG 2544, Döllnsee, September 16, 2020.

W. König, Titel folgt (kein Archiv auf HP), Oberseminar Mathematische Stochastik, Westfälischen Wilhelms-Universität Münster, Fachbereich Mathematik und Informatik, January 22, 2020.

L. Taggi, Macroscopic self-avoiding walk interacting with lattice permutations and uniformly-positive monomer-correlations for the dimer model in $Z^d, d > 2$, Probability Seminar, University of Warwick, Mathematics Institute, UK, January 16, 2020.

L. Taggi, Macroscopic self-avoiding walk interacting with lattice permutations and uniformly-positive monomer-monomer correlations in the dimer model in $Z^d, d > 2$, Probability Seminar, University of Bristol, School of Mathematics Research, France, April 7, 2020.

L. Taggi, Macroscopic self-avoiding walk interacting with lattice permutations and uniformly-positive monomer-monomer correlations in the dimer model in $Z^d, d > 2$, Probability Seminar, University of Bristol, School of Mathematics, UK, February 7, 2020.

M. Coghi, Mean field limit of interacting filaments for 3D Euler equations, Second Italian Meeting on Probability and Mathematical Statistics, June 17 - 20, 2019, Università degli Studi di Salerno, Dipartimento di Matematica, Vietri sul Mare, Italy, June 20, 2019.

A. Hinsen, Data mobility in ad-hoc networks: Vulnerability and security, KEIN öffentlicher Vortrag (Orange), Telecom Orange Paris, France, December 12, 2019.

A. Hinsen, IPS in telecommunication I, Workshop on Probability, Analysis and Applications (PAA), September 23 - October 4, 2019, African Institute for Mathematical Sciences --- Ghana (AIMS Ghana), Accra, October 4, 2019.

A. Hinsen, IPS in telecommunication II, Workshop on Probability, Analysis and Applications (PAA), September 23 - October 4, 2019, African Institute for Mathematical Sciences --- Ghana (AIMS Ghana), Accra, October 4, 2019.

A. Hinsen, Introduction to interacting particles systems (IPS), Workshop on Probability, Analysis and Applications (PAA), September 23 - October 4, 2019, African Institute for Mathematical Sciences --- Ghana (AIMS Ghana), Accra, October 2, 2019.

A. Hinsen, The White Knight model --- An epidemic on a spatial random network, Bocconi Summer School in Advanced Statistics and Probability, Lake Como School of Advanced Studies, Lake Como, Italy, July 8 - 19, 2019.

A. Hinsen, Typical Voronoi cells for Cox point processes on Manhattan grids, The International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt 2019) [The 13th Workshop on Spatial Stochastic Models for Wireless Networks (SpasWin 2019 )], June 7, 2019, Avignon, France, June 7, 2019.

T. Orenshtein, Random walks in random environment as rough paths, Probability Seminar, New York University Shanghai, Institute of Mathematical Sciences, Shanghai, China, November 26, 2019.

D.R.M. Renger, A generic formulation of a chemical reaction network from Onsager--Machlup theory, Conference to Celebrate 80th Jubilee of Miroslav Grmela, May 18 - 19, 2019, Czech Technical University, Faculty of Nuclear Sciences and Physical Engineering, Prague, May 19, 2019.

D.R.M. Renger, Macroscopic fluctuation theory of chemical reaction networks, Workshop on Chemical Reaction Networks, July 1 - 3, 2019, Politecnico di Torino, Dipartimento di Scienze Matematiche ``G. L. Lagrange``, Italy, July 2, 2019.

D.R.M. Renger, Reaction fluxes, Applied Mathematics Seminar, University of Birmingham, School of Mathematics, UK, April 4, 2019.

W. van Zuijlen, Bochner integrals in ordered vector spaces, Analysis Seminar, University of Canterbury, Department of Mathematics and Statistics, UK, March 1, 2019.

W. van Zuijlen, From periodic to Dirichlet and Neumann on boxes, 12th Oxford-Berlin Young Researchers Meeting on Applied Stochastic Analysis, December 4 - 6, 2019, University of Oxford, Mathematical Institute, UK, December 6, 2019.

W. van Zuijlen, From periodic to Dirichlet and Neumann on boxes, Seminar Forschergruppe 2402: Research Unit --- Rough paths, stochastic partial differential equations and related topics, Technische Universität Berlin, Institut für Mathematik, December 12, 2019.

W. van Zuijlen, Mass-asymptotics for the parabolic Anderson model in 2D, Berlin--Leipzig Workshop in Analysis and Stochastics, January 16 - 18, 2019, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, January 18, 2019.

W. van Zuijlen, Mini-course on Besov spaces I--III, Junior Trimester Program: Randomness, PDEs and Nonlinear Fluctuations (Sept. 2 to Dec. 19, 2019), October 16 - November 6, 2019, Hausdorff Research Institute for Mathematics (HIM), Bonn.

W. van Zuijlen, The parabolic Anderson model in 2D, mass- and eigenvalue asymptotics, Stochastic Analysis Seminar, University of Oxford, Mathematical Institute, UK, February 4, 2019.

W. van Zuijlen, The parabolic Anderson model in 2D, mass- and eigenvalue asymptotics, Analysis and Probability Seminar, Imperial College London, Department of Mathematics, UK, February 5, 2019.

L. Andreis, A large-deviations approach to the multiplicative coagulation process, Workshop ``Woman in Probability'', May 31 - June 1, 2019, Technische Universität München, Fakultät für Mathematik, May 31, 2019.

L. Andreis, Coagulating particles and gelation phase transition: A large-deviation approach, Second Italian Meeting on Probability and Mathematical Statistics, June 17 - 20, 2019, Vietri sul Mare, Italy, June 19, 2019.

L. Andreis, Coagulation processes and gelation from a large deviation point of view, BMS -- BGSMath Junior Meeting 2019, June 26 - 28, 2019, Berlin Mathematical School (BMS), Barcelona Graduate School of Mathematics (BGSMath), Technische Universität Berlin, June 26, 2019.

L. Andreis, Large-deviation approach to coagulation processes and gelation, Workshop on Chemical Reaction Networks, July 1 - 3, 2019, Politecnico di Torino, Dipartimento di Scienze Matematiche ''G. L. Lagrange``, Italy, July 2, 2019.

L. Andreis, Multiplicative coagulation process and random graphs in sparse regime: a large-deviations approach, STAR Workshop on Random Graphs 2019, April 10 - 12, 2019, University of Groningen, Department of Mathematics and Natural Sciences, Netherlands, April 12, 2019.

L. Andreis, Phase transitions in coagulation processes and random graphs, Workshop ``Welcome Home 2019'', December 19 - 20, 2019, Università di Torino, Dipartimento di Matematica ``G. Peano'', Italy, December 19, 2019.

B. Jahnel, Attractor properties for irreversible and reversible interacting particle systems, Quaid-i-Azam University Islamabad, Department of Mathematics, Pakistan, November 19, 2019.

B. Jahnel, Continuum percolation in random environment, Workshop on Probability, Analysis and Applications (PAA), September 23 - October 4, 2019, African Institute for Mathematical Sciences --- Ghana (AIMS Ghana), Accra.

B. Jahnel, Coverage and mobility in device-to-device networks, Kein öffentlicher Vortrag (Orange), Telecom Orange Paris, France, December 12, 2019.

B. Jahnel, Dynamical Gibbs-non-Gibbs transitions for the continuum Widom--Rowlinson model, The 41st Conference on Stochastic Processes and their Applications 2019 (SPA 2019), July 8 - 12, 2019, Northwestern University Evanston, USA, July 9, 2019.

B. Jahnel, Is the Mathern process Gibbs?, Workshop on Stochastic Modeling of Complex Systems, GWOT '19, April 8 - 12, 2019, Universität Mannheim, Institut für Mathematik, April 9, 2019.

B. Jahnel, Three models for data propagation in mobile ad-hoc networks, kein öffentlicher Vortrag (Orange), Telecom Orange Paris, France, December 12, 2019.

W. König, A large-deviations approach to coagulation, Workshop on Stochastic Modeling of Complex Systems, GWOT '19, April 8 - 12, 2019, Universität Mannheim, Institut für Mathematik, April 10, 2019.

W. König, A large-deviations approach to coagulation, Maxwell Analysis Seminar, Harriot Watt University, The Maxwell Institute for Mathematical Sciences, Edinburgh, UK, December 27, 2019.

W. König, A large-deviations approach to the multiplicative coalescent, Math Probability Seminar Series, New York University Shanghai, Institute of Mathematical Sciences, China, February 19, 2019.

W. König, A large-deviations approach to the multiplicative coalescent, 18. Erlanger-Münchner Tag der Stochastik / Probability Day 2019, Friedrich-Alexander-Universität Erlangen--Nürnberg, Department Mathematik, May 10, 2019.

W. König, A large-deviations approach to the multiplicative coalescent, Oberseminar Wahrscheinlichkeitstheorie, Universität München, Mathematisches Institut, November 25, 2019.

W. König, Cluster size distribution in a classical many-body system, Deutsches Zentrum für Luft- und Raumfahrt (DLR), Institut für Materialphysik im Weltraum, Köln, June 18, 2019.

W. König, EF4: Particles and Agents, 1st MATH+ Day, Berlin, December 13, 2019.

W. König, EF4: Particles and Agents, MATH+ Day, URANIA, December 13, 2019.

W. König, Eigenvalue order statistics and mass concentration in the parabolic Anderson model, Berlin--Leipzig Workshop in Analysis and Stochastics, January 16 - 18, 2019, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, January 17, 2019.

W. König, Eigenvalue order statistics and mass concentration in the parabolic Anderson model, Workshop on Spectral Properties of Disordered Systems, January 7 - 11, 2019, Paris, France, January 11, 2019.

W. König, Micro-macro phase transitions in coagulating particle systems, Workshop on Probability, Analysis and Applications (PAA), September 23 - October 4, 2019, African Institute for Mathematical Sciences --- Ghana (AIMS Ghana), Accra.

R.I.A. Patterson, A novel simulation method for stochastic particle systems, Seminar, Department of Chemical Engineering and Biotechnology, University of Cambridge, Faculty of Mathematics, UK, May 9, 2019.

R.I.A. Patterson, Fluctuations and confidence intervals for stochastic particle simulations, First Berlin--Leipzig Workshop on Fluctuating Hydrodynamics, August 26 - 30, 2019, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, August 29, 2019.

R.I.A. Patterson, Flux large deviations, Workshop on Chemical Reaction Networks, July 1 - 3, 2019, Politecnico di Torino, Dipartimento di Scienze Matematiche ``G. L. Lagrange``, Italy, July 2, 2019.

R.I.A. Patterson, Flux large deviations, Seminar, Statistical Laboratory, University of Cambridge, Faculty of Mathematics, UK, May 7, 2019.

R.I.A. Patterson, Interaction clusters for the Kac process, Berlin--Leipzig Workshop in Analysis and Stochastics, January 16 - 18, 2019, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, January 18, 2019.

R.I.A. Patterson, Interaction clusters for the Kac process, Workshop on Effective Equations: Frontiers in Classical and Quantum Systems, June 24 - 28, 2019, Hausdorff Research Institute for Mathematics, Bonn, June 28, 2019.

R.I.A. Patterson, Kinetic interaction clusters, Oberseminar, Martin-Luther-Universität Halle-Wittenberg, Naturwissenschaftliche Fakultät II -- Chemie, Physik und Mathematik, April 17, 2019.

R.I.A. Patterson, The role of fluctuating hydrodynamics in the CRC 1114, CRC 1114 School 2019: Fluctuating Hydrodynamics, Zuse Institute Berlin (ZIB), October 28, 2019.

L. Taggi, Absorbing-state phase transition in activated random walk and oil and water, Probability Seminars, Università degli Studi ``La Sapienza'' di Roma, Italy, October 1, 2019.

L. Taggi, Critical density in activated random walks, Horowitz Seminar on Probability, Ergodic Theory and Dynamical Systems, Tel Aviv University, School of Mathematical Sciences, Israel, May 20, 2019.

L. Taggi, Essential enhancements for activated random walks, Second Italian Meeting on Probability and Mathematical Statistics, June 17 - 20, 2019, Vietri sul Mare, Italy, June 19, 2019.

L. Taggi, Non-decay of correlations in the dimer model and phase transition in lattice permutations in $Z^d, d > 2$ via reflection positivity, Meeting of the Swiss Mathematical Society: Recent Advances in Loop Models and Height Functions, September 2 - 4, 2019, Université Fribourg, Switzerland, September 3, 2019.

L. Taggi, Phase transition in lattice permutations and uniformly positive correlations in the dimer model in $Z^d, d > 2$, via reflection positivity, Kolloquium des Fachbereichs Mathematik, Technische Universität Darmstadt, December 5, 2019.

L. Taggi, Uniformly positive correlations in the dimer model and phase transition in lattice permutations in $Z^d, d > 2$, via reflection positivity, Seminaire Probabilités et Statistiques, Université Claude Bernard Lyon 1, Institut Camille Jordan (ICJ), France, November 14, 2019.

A. Hinsen, Random Malware Propagation, MATH+ Center Days 2018, October 31 - November 2, 2018, Zuse-Institut Berlin (ZIB), Berlin, October 31, 2018.

A. Hinsen, The White Knight Model - Propagation of Malware on a D2D Network, 14. Doktorand*innentreffen Stochastik, Essen 2018, August 1 - 3, 2018, Universität Duisburg--Essen, August 3, 2018.

A. Hinsen, Vulnerability and security in ad-hoc networks, Universität Osnabrück, Fachbereich Mathematik/Informatik, December 11, 2018.

M. Maurelli, Sanov theorem for Brownian rough paths and an application to interacting particles, Università di Roma La Sapienza, Dipartimento di Matematica Guido Castelnuovo, Italy, January 18, 2018.

D.R.M. Renger, Gradient and GENERIC structures from flux large deviations, POLYPHYS Seminar, Eidgenössische Technische Hochschule Zürich, Department of Materials, Zürich, Switzerland, March 28, 2018.

D.R.M. Renger, Gradient and GENERIC structures in the space of fluxes, Analysis of Evolutionary and Complex Systems (ALEX2018), September 24 - 28, 2018, WIAS Berlin, September 27, 2018.

D.R.M. Renger, Gradient and Generic structures in the space of fluxes, Analysis of Evolutionary and Complex Systems (ALEX2018), September 24 - 28, 2018, WIAS Berlin, September 27, 2018.

D.R.M. Renger, Large deviations for reaction fluxes, Workshop on Transformations and Phase Transitions, January 29 - 31, 2018, Ruhr-Universität Bochum, Fakultät für Mathematik, January 29, 2018.

D.R.M. Renger, Large deviations for reaction fluxes, Università degli Studi dell'Aquila, Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica, L'Aquila, Italy, January 10, 2018.

R. Soares Dos Santos, Random walk on random walks, University of Groningen, Johann Bernoulli Institute for Mathematics and Computer Science, Netherlands, February 14, 2018.

R. Soares Dos Santos, Random walk on random walks, Oberseminar Mathematische Stochastik, Westfälische Wilhelms-Universität Münster, Fachbereich Mathematik und Informatik, Münster, July 4, 2018.

R. Soares Dos Santos, The parabolic Anderson model with renormalized inverse-square Poisson potential, 13th German Probability and Statistics Days 2018 -- Freiburger Stochastik-Tage, February 27 - March 2, 2018, Albert-Ludwigs-Universität Freiburg, Abteilung für Mathematische Stochastik, Freiburg, February 2, 2018.

W. van Zuijlen, A Hamilton--Jacobi point of view on mean-field Gibbs-non-Gibbs transitions, Workshop on Transformations and Phase Transitions, January 29 - 31, 2018, Ruhr-Universität Bochum, Fakultät für Mathematik, Bochum, January 30, 2018.

W. van Zuijlen, Eigenvalues of the Anderson Hamiltonian with white noise potential in 2D, Leiden University, Institute of Mathematics, Leiden, Netherlands, May 1, 2018.

W. van Zuijlen, Mass-asymptotics for the parabolic Anderson model in 2D, 10th Oxford-Berlin Young Researchers Meeting on Applied Stochastic Analysis, November 29 - December 1, 2018, University of Oxford, Mathematical Institute, Oxford, UK, November 29, 2018.

W. van Zuijlen, Mass-asymptotics for the parabolic Anderson model in 2D, Statistical Mechanics Seminar, University of Warwick, Department of Statistics, Coventry, UK, December 6, 2018.

W. van Zuijlen, Mean-field Gibbs-non-Gibbs transitions, Spring School, Spin Systems: Discrete and Continuous, March 19 - 23, 2018, Technische Universität Darmstadt, Fachbereich Mathematik, Darmstadt.

W. van Zuijlen, The principal eigenvalue of the Anderson Hamiltonian in continuous space, 13th German Probability and Statistics Days 2018 -- Freiburger Stochastik-Tage, February 27 - March 2, 2018, Albert-Ludwigs-Universität Freiburg, Abteilung für Mathematische Stochastik, Freiburg, February 28, 2018.

L. Andreis, A large-deviations approach to the multiplicative coagulation process, Probability Seminar, Università degli Studi di Padova, Dipartimento di Matematica ``Tullio Levi--Civita'', Italy, October 12, 2018.

L. Andreis, A large-deviations approach to the multiplicative coagulation process, Seminar ''Theory of Complex Systems and Neurophysics --- Theory of Statistical Physics and Nonlinear Dynamics``, Humboldt-Universität zu Berlin, Institut für Physik, October 30, 2018.

L. Andreis, Ergodicity of a system of interacting random walks with asymmetric interaction, 13th German Probability and Statistics Days 2018 -- Freiburger Stochastik-Tage, February 27 - March 2, 2018, Albert-Ludwigs-Universität Freiburg, Abteilung für Mathematische Stochastik, Freiburg, February 1, 2018.

L. Andreis, Networks of interacting components with macroscopic self-sustained periodic behavior, Neural Coding 2018, September 9 - 14, 2018, University of Torino, Department of Mathematics, Italy, September 10, 2018.

L. Andreis, Self-sustained periodic behavior in interacting systems, Random Structures in Neuroscience and Biology, March 26 - 29, 2018, Ludwig--Maximilians Universität München, Fakultät für Mathematik, Informatik und Statistik, Herrsching, March 26, 2018.

L. Andreis, System of interacting random walks with asymmetric interaction, 48th Probability Summer School, July 8 - 20, 2018, Clermont Auvergne University, Saint Flour, France, July 17, 2018.

F. Flegel, Localization vs. homogenization in the random conductance model, Forschungsseminar Analysis, Technische Universität Chemnitz, June 6, 2018.

F. Flegel, Spectral homogenization vs. localization in the barrier model, Symposium on the occasion of the 60th birthday of Igor Sokolov, Bernstein Center for Computational Neuroscience Berlin, Humboldt-Universität zu Berlin, February 26, 2018.

F. Flegel, Spectral homogenization vs. localization in the random conductance model, Workshop `` Interplay of Analysis and Probability in Applied Mathematics'', February 11 - 17, 2018, Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach, February 12, 2018.

F. Flegel, Spectral homogenization vs. localization in the random conductance model, Seminar Angewandte Analysis, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, March 9, 2018.

F. Flegel, Spectral localization vs. homogenization in the random conductance model, Seminar of SFB/CRC 1060 Bonn, Rheinische Friedrich-Wilhelms-Universität Bonn, June 12, 2018.

F. Flegel, Spectral localization vs. homogenization in the random conductance model, Oberseminar Stochastik, Universität zu Köln, Mathematisches Institut, June 14, 2018.

F. Flegel, Spectral localization vs. homogenization in the random conductance model, Oberseminar Wahrscheinlichkeitstheorie, Ludwigs-Maximilians-Universität München, July 9, 2018.

B. Jahnel, Attractor properties for irreversible and reversible interacting particle systems, ICM 2018 Satellite Conference: Topics in Mathematical Physics, July 26 - 31, 2018, Institute of Mathematics and Statistics, University of São Paulo, Institute of Physics, Brazil, July 27, 2018.

B. Jahnel, Attractor properties for irreversible and reversible interacting particle systems, Random Structures in Neuroscience and Biology, March 26 - 29, 2018, Ludwig--Maximilians Universität München, Fakultät für Mathematik, Informatik und Statistik, Herrsching, March 29, 2018.

B. Jahnel, Attractor properties for irreversible and reversible interacting particle systems, Ibn Zohr University, Agadir, Morocco, September 28, 2018.

B. Jahnel, Continuum percolation for Cox point processes, 13th German Probability and Statistics Days 2018 -- Freiburger Stochastik-Tage, February 27 - March 2, 2018, Albert-Ludwigs-Universität Freiburg, Abteilung für Mathematische Stochastik, Freiburg, February 28, 2018.

B. Jahnel, Continuum percolation for Cox point processes, Seminar, Universität Potsdam, Institut für Mathematik, April 13, 2018.

B. Jahnel, Continuum percolation for Cox point processes, Universität Osnabrück, Fachbereich Mathematik/Informatik, February 1, 2018.

B. Jahnel, Dynamical Gibbs-non-Gibbs transitions for continuous spin models, DFG-AIMS Workshop on Evolutionary Processes on Networks, March 20 - 24, 2018, African Institute of Mathematical Sciences (AIMS), Kigali, Rwanda, March 21, 2018.

B. Jahnel, Dynamical Gibbs-non-Gibbs transitions for the continuum Widom--Rowlinson model, Seminar der AG Stochastik, Technische Universität Darmstadt, Fachbereich Mathematik, September 21, 2018.

B. Jahnel, Gibbsian representation for point processes via hyperedge potentials, Workshop on Transformations and Phase Transitions, January 29 - 31, 2018, Ruhr-Universität Bochum, Fakultät für Mathematik, Bochum, January 30, 2018.

B. Jahnel, Gibbsian representation for point processes via hyperedge potentials, Universität Potsdam, Institut für Mathematik, October 10, 2018.

B. Jahnel, Influence of mobility on connectivity, MATH+ Center Days 2018, October 31 - November 2, 2018, Zuse-Institut Berlin (ZIB), Berlin, October 31, 2018.

B. Jahnel, Percolation for Cox point processes, Geometry and Scaling of Random Structures, July 16 - 27, 2018, Centre International de Mathématiques Pures et Appliquées (CIMPA), School and X Escuela Santaló, ICM Rio Satellite Workshop, Buenos Aires, Argentina, July 18, 2018.

B. Jahnel, Spatial stochastic models with applications in telecommunications, Summer School 2018 ``Combinatorial Structures in Geometry'', September 24 - 27, 2018, Universität Osnabrück, Institut für Mathematik (DFG GK 1916).

B. Jahnel, Telecommunication models in random environments, BIMoS Day : The Mathematics of Quantum Information, May 23, 2018, Technische Universität Berlin, Berlin, May 23, 2018.

B. Jahnel, Telecommunication models in random environments, BIMoS Day, Berlin International Graduate School in Model and Simulation based Research, Technische Universität Berlin, May 23, 2018.

W. König, A large-deviations approach to the multiplicative coalescent, Workshop on High-dimensional Phenomena in Probability --- Fluctuations and Discontinuity (Research Training Group 2131), September 24 - 28, 2018, Ruhr-Universität Bochum, September 28, 2018.

W. König, Large deviations theory and applications, Classical and Quantum Dynamics of Interacting Particle Systems, June 15, 2018, Universität zu Köln, Mathematisches Institut, Köln.

W. König, Probabilistic Methods in Telecommunication, MATH+ Center Days 2018, October 31 - November 2, 2018, Zuse-Institut Berlin (ZIB), Berlin, October 31, 2018.

W. König, Random message routing in highly dense multi-hop networks, DFG-AIMS Workshop on Evolutionary Processes on Networks, March 20 - 24, 2018, African Institute of Mathematical Sciences (AIMS), Kigali, Rwanda, March 21, 2018.

W. König, The principal part of the spectrum of random Schrödinger operators in large boxes, Rhein-Main Kolloquium Stochastik, Goethe-Universität Frankfurt am Main, January 26, 2018.

R.I.A. Patterson, Large deviations for reaction fluxes, Séminaire EDP, Modélisation et Calcul Scientifique (commun ICJ & UMPA), École Normale Supérieure de Lyon (CNRS), France, July 12, 2018.

R.I.A. Patterson, Large deviations for reaction fluxes, Workshop `` Interplay of Analysis and Probability in Applied Mathematics'', February 11 - 17, 2018, Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach, February 15, 2018.

D.R.M. Renger, Gradient flows and GENERIC in flux space, Workshop ``Variational Methods for Evolution'', November 12 - 18, 2017, Mathematisches Forschungsinstitut Oberwolfach, November 16, 2017.

A. González Casanova Soberón, Branching processes with interactions and their relation to population genetics, The 3rd Workshop on Branching Processes and Related Topics, May 8 - 12, 2017, Beijing Normal University, School of Mathematical Sciences, China, May 8, 2017.

A. González Casanova Soberón, Modeling selection via multiple parents, Annual Colloquium SPP 1590, October 4 - 6, 2017, Albert-Ludwigs-Universität Freiburg, Fakultät für Mathematik und Physik, October 6, 2017.

A. González Casanova Soberón, Modelling selection via multiple parents, Seminar Probability, National Autonomous University of Mexico, Mexico City, February 23, 2017.

A. González Casanova Soberón, Modelling selection via multiple parents, Probability Seminar, University of Oxford, Mathematical Institute, UK, January 24, 2017.

A. González Casanova Soberón, Modelling the Lenski experiment, 19th ÖMG Congress and Annual DMV Meeting, Section S16 ``Mathematics in the Science and Technology'', September 11 - 15, 2017, Austrian Mathematical Society (ÖMG) and Deutsche Mathematiker-Vereinigung (DMV), Paris-Lodron University of Salzburg, Austria, September 14, 2017.

A. González Casanova Soberón, The ancestral efficiency graph, Spatial Models in Population Genetics, September 6 - 8, 2017, University of Bath, Department of Mathematical Sciences, UK, September 6, 2017.

A. González Casanova Soberón, The discrete ancestral selection graph, Seminar, Center for Interdisciplinary Research in Biology, Stochastic Models for the Inference of Life Evolution SMILE, Paris, France, October 23, 2017.

B. Jahnel, The Widom--Rowlinson model under spin flip: Immediate loss and sharp recovery of quasilocality, Oberseminar Wahrscheinlichkeitstheorie, Ludwig-Maximilians-Universität München, Fakultät für Mathematik, Informatik und Statistik, February 13, 2017.

B. Jahnel, The Widom--Rowlinson model under spin flip: Immediate loss and sharp recovery of quasilocality, Westfälische Wilhelms-Universität Münster, Fachbereich Mathematik und Informatik, January 18, 2017.

B. Jahnel, The Widom--Rowlinson model under spin flip: Immediate loss and sharp recovery of quasilocality, Oberseminar Stochastik, Johannes Gutenberg Universität Mainz, Institut für Mathematik, April 25, 2017.

B. Jahnel, The Widom-Rowlinson model under spin flip: immediate loss and sharp recovery of quasilocality, Université du Luxembourg, Faculté des Sciences, de la Technologie et de la Communication (FSTC), Luxembourg, March 3, 2017.

P. Keeler, Optimizing spatial throughput in device-to-device networks, Applied Probability @ The Rock -- An International Workshop celebrating Phil Pollett's 60th Birthday, April 17 - 21, 2017, University of Adelaide, School of Mathematical Sciences, Uluru, Australia, April 20, 2017.

CH. Mukherjee, Asymptotic behavior of the mean-field polaron, Probability and Mathematical Physics Seminar, Courant Institute of Mathematical Sciences, Department of Mathematics, New York, USA, March 20, 2017.

R.I.A. Patterson, Confidence intervals for coagulation--advection simulations, Clausthal-Göttingen International Workshop on Simulation Science, April 27 - 28, 2017, Georg-August-Universität Göttingen, Institut für Informatik, April 28, 2017.

D.R.M. Renger, Large deviations and gradient flows, Spring School 2017: From Particle Dynamics to Gradient Flows, February 27 - March 3, 2017, Technische Universität Kaiserslautern, Fachbereich Mathematik, March 1, 2017.

D.R.M. Renger, Was sind und was sollen die Zahlen, Tag der Mathematik, Universität Regensburg, Fakultät für Mathematik, July 28, 2017.

R. Soares Dos Santos, Complete localisation in the Bouchaud--Anderson model, Leiden University, Institute of Mathematics, Netherlands, May 9, 2017.

R. Soares Dos Santos, Concentration de masse dans le modèle parabolique d'Anderson, Séminaire de Probabilités, Université de Grenoble, Institut Fourier, Laboratoire des Mathematiques, France, April 11, 2017.

R. Soares Dos Santos, Eigenvalue order statistics of random Schrödinger operators and applications to the parabolic Anderson model, 19th ÖMG Congress and Annual DMV Meeting, Minisymposium M6 ``Spectral and Scattering Problems in Mathematical Physics'', September 11 - 15, 2017, Austrian Mathematical Society (ÖMG) and Deutsche Mathematiker-Vereinigung (DMV), Paris-Lodron University of Salzburg, Austria, September 12, 2017.

W. van Zuijlen, Mean-field Gibbs-non-Gibbs transitions, Mark Kac Seminar, Utrecht University, Mathematical Institute, Netherlands, February 3, 2017.

W. van Zuijlen, The principal eigenvalue of the Anderson Hamiltonian in continuous space, Berlin-Leipzig Workshop in Analysis and Stochastics, November 29 - December 1, 2017, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, November 29, 2017.

L. Andreis, McKean-Vlasov limits, propagation of chaos and long-time behavior of some mean field interacting particle systems, Verteidigung Dissertation, November 15 - 20, 2017, Università degli Studi di Padova, Dipartimento di Matematica Pura ed Applicata, Padova, Italy, November 16, 2017.

F. Flegel, Spectral localization vs. homogenization in the random conductance model, 19th ÖMG Congress and Annual DMV Meeting, Minisymposium M6 ``Spectral and Scattering Problems in Mathematical Physics'', September 11 - 15, 2017, Austrian Mathematical Society (ÖMG) and Deutsche Mathematiker-Vereinigung (DMV), Paris-Lodron University of Salzburg, Austria, September 12, 2017.

F. Flegel, Spectral localization vs. homogenization in the random conductance model, Berlin-Leipzig Workshop in Analysis and Stochastics, November 29 - December 1, 2017, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, November 29, 2017.

B. Jahnel, Gibbsian representation for point processes via hyperedge potentials, Workshop on Stochastic Analysis and Random Fields, Second Haifa Probability School, December 18 - 22, 2017, Technion Israel Institute of Technology, Haifa, Israel, December 18, 2017.

B. Jahnel, Large deviations in relay-augmented wireless networks, Sharif University of Technology Tehran, Mathematical Sciences Department, Teheran, Iran, September 17, 2017.

W. König, A variational formula for an interacting many-body system, Probability Seminar, University of California, Los Angeles, Department of Mathematics, USA, January 19, 2017.

W. König, Cluster-size distributions in a classical many-body system, Berlin-Leipzig Workshop in Analysis and Stochastics, November 29 - December 1, 2017, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, November 29, 2017.

W. König, Connectivity in large mobile ad-hoc networks, Summer School 2017: Probabilistic and Statistical Methods for Networks, August 21 - September 1, 2017, Technische Universität Berlin, Berlin Mathematical School, August 29, 2017.

W. König, Intersections of Brownian motions, Workshop ``Peter's Network'', October 31 - November 1, 2017, University of Bath, Department of Mathematical Sciences, UK, November 1, 2017.

W. König, Moment asymptotics of branching random walks in random environment, Modern Perspective of Branching in Probability, September 26 - 29, 2017, Westfälische Wilhelms-Universität Münster, Fachbereich Mathematik und Informatik, September 28, 2017.

W. König, The principal part of the spectrum of a random Schrödinger operator in a large box, Mathematisches Kolloquium, Oberseminar Stochastik und Analysis, Technische Universität Dormund, May 15, 2017.

R.I.A. Patterson, Coagulation --- Transport Simulations with Stochastic Particles, CIM-WIAS Workshop ``Topics in Applied Analysis and Optimisation'', December 6 - 8, 2017, University of Lisbon, International Center for Mathematics, Lisboa, Portugal, December 7, 2017.

R.I.A. Patterson, Simulation of particle coagulation and advection, Numerical Methods and Applications of Population Balance Equations, October 13, 2017, GRK 1932, Technische Universität Kaiserslautern, Fachbereich Mathematik, October 13, 2017.

A. Pandey, Meshfree method for fluctuating hydrodynamics, International Conference on Advances in Scientific Computing, November 28 - 30, 2016, Indian Institute of Technology, Department of Mathematics, Madras, November 30, 2016.

A. González Casanova Soberón, An individual based model for the Lenski experiment, 1st Leibniz MMS Days, January 27 - 29, 2016, WIAS Berlin, Berlin, January 27, 2016.

A. González Casanova Soberón, An individual based model for the Lenski experiment, 1st Leibniz MMS Days, WIAS Berlin, Berlin, January 27, 2016.

A. González Casanova Soberón, Fixation in a Xi coalescent model with selection, Probability seminar, University of Warwick, Mathematics Institute, Warwick, UK, November 30, 2016.

A. González Casanova Soberón, Fixation in a Xi coalescent with selection, Miniworkshop on Probabilistic Models in Evolutionary Biology, November 24 - 25, 2016, Georg-August-Universität Göttingen, Institut für Mathematische Stochastik, November 25, 2016.

A. González Casanova Soberón, Modeling the Lenski experiment, Mathematical and Computational Evolutionary Biology, June 12 - 16, 2016, Le Laboratoire d'Informatique, de Robotique et de Microélectronique de Montpellier (LIRMM), Hameau de l'Etoile, France, June 14, 2016.

A. González Casanova Soberón, The seed bank model, VIII School on Probability and Stochastic Processes, September 12 - 16, 2016, Centro de Investigación en Matemáticas (CIMAT), Department of Probability and Statistics, Guanajuato, Mexico.

A. González Casanova Soberón, The seedbank coalescent, World Congress in Probability and Statistics, Invited Session ``Stochastic Models of Evolution'', July 11 - 15, 2016, Fields Institute, Toronto, Canada, July 5, 2016.

CH. Hirsch, From heavy-tailed Boolean models to scale-free Gilbert graphs, Workshop on Continuum Percolation, January 26 - 29, 2016, University Lille 1, Science et Technologies, France, January 28, 2016.

CH. Hirsch, Large deviations in relay-augmented wireless networks, Workshop on Dynamical Networks and Network Dynamics, January 17 - 22, 2016, International Centre for Mathematical Science, Edinburgh, UK, January 18, 2016.

CH. Hirsch, Large deviations in relay-augmented wireless networks, 12th German Probability and Statistics Days 2016 -- Bochumer Stochastik-Tage, February 29 - March 4, 2016, Ruhr-Universität Bochum, Fakultät für Mathematik, March 3, 2016.

CH. Hirsch, On maximal hard-core thinnings of stationary particle processes, Oberseminar Wahrscheinlichkeitstheorie, Ludwig-Maximilians-Universität München, Fakultät für Mathematik, April 18, 2016.

B. Jahnel, Attractor properties for irreversible and reversible interacting particle systems, 12th German Probability and Statistics Days 2016 -- Bochumer Stochastik-Tage, February 29 - March 4, 2016, Ruhr-Universität Bochum, Fakultät für Mathematik, March 3, 2016.

B. Jahnel, Classes of nonergodic interacting particle systems with unique invariant measure, Romanian Academy of Sciences, Institute of Mathematical Statistics and Applied Mathematics, Bucharest, February 22, 2016.

B. Jahnel, GnG transitions for the continuum Widom--Rowlinson model under spin flip: Immediate loss and sharp recovery of quasilocality, Transformations in Statistical Mechanics: Pathologies and Remedies, October 9 - 14, 2016, Lorentz Center -- International Center for Scientific Workshops, Leiden, Netherlands, October 11, 2016.

P. Keeler, Signal-to-interference ratio in wireless communication networks, Workshop on Dynamical Networks and Network Dynamics, January 17 - 24, 2016, International Centre for Mathematical Science, Edinburgh, UK, January 18, 2016.

P. Keeler, Wireless network models: Geometry OR signal-to-interference ratio (SIR), Paris, Paris, France, June 3, 2016.

M. Maurelli, Enhanced Sanov theorem and large deviations for interacting particles, Workshop ``Rough Paths, Regularity Structures and Related Topics'', May 1 - 7, 2016, Mathematisches Forschungsinstitut Oberwolfach, May 5, 2016.

CH. Mukherjee, Compactness and large deviations, Probability Seminar, Stanford University, Department of Mathematics and Statistics, USA, November 14, 2016.

CH. Mukherjee, Compactness and large deviations, Mathematisches Kolloquium, Universität Konstanz, Fachbereich Mathematik und Statistik, May 18, 2016.

CH. Mukherjee, Compactness and large deviations, Probability Seminar, University of California at Berkeley, Department of Statistics, USA, October 19, 2016.

CH. Mukherjee, Compactness, large deviations and statistical mechanics, Seminar des Fachbereichs Mathematik und Statistik, Westfälische Wilhelms-Universität Münster, Fachbereich Mathematik und Informatik, October 17, 2016.

CH. Mukherjee, Compactness, large deviations and the polaron, Probability Seminar, University of Washington, Department of Mathematics, Seattle, USA, October 31, 2016.

CH. Mukherjee, Compactness, large deviations and the polaron, The University of Arizona, Department of Mathematics, USA, November 2, 2016.

CH. Mukherjee, Compactness, large deviations, and the polaron problem, 12th German Probability and Statistics Days 2016 -- Bochumer Stochastik-Tage, February 29 - March 4, 2016, Ruhr-Universität Bochum, Fakultät für Mathematik, March 3, 2016.

CH. Mukherjee, Occupation measures, compactness and large deviations, Young European Probabilists Workshop ``Large Deviations for Interacting Particle Systems and Partial Differential Equations'' (YEP XIII), March 6 - 11, 2016, European Institute for Statistics, Probability, Stochastic Operations Research and their Applications (EURANDOM), Eindhoven, Netherlands, March 7, 2016.

CH. Mukherjee, On some aspects of large deviations, Mathematics Colloquium, West Virginia University, Department of Mathematics, Morgantown, USA, March 17, 2016.

CH. Mukherjee, Polaron problem, Probability Seminar, University of California at Irvine, Department of Mathematics, USA, October 25, 2016.

CH. Mukherjee, Quenched large deviations for random walks on supercritical percolation clusters, Probability and Mathematical Physics Seminar, Courant Institute, New York, Department of Mathematics, USA, November 4, 2016.

CH. Mukherjee, The polaron problem, Rutgers University, Department of Mathematics, New Brunswick, USA, November 17, 2016.

CH. Mukherjee, Weak/strong disorder for stochastic heat equation, Analysis Seminar, University of California at Berkeley, Department of Mathematics, USA, October 21, 2016.

CH. Mukherjee, Weak/strong disorder for stochastic heat equation, Probability and Mathematical Physics Seminar, University of California at Los Angeles, Department of Mathematics, USA, October 27, 2016.

CH. Mukherjee, Weak/strong disorder for stochastic heat equation, City University of New York, Department of Mathematics, USA, November 8, 2016.

D.R.M. Renger, Functions of bounded variation with an infinite-dimensional codomain, Meeting in Applied Mathematics and Calculus of Variations, September 13 - 16, 2016, Università di Roma ``La Sapienza'', Dipartimento di Matematica ``Guido Castelnuovo'', Italy, September 16, 2016.

D.R.M. Renger, Large deviations for reacting particle systems: The empirical and ensemble process, Young European Probabilists Workshop ``Large Deviations for Interacting Particle Systems and Partial Differential Equations'' (YEP XIII), March 6 - 11, 2016, European Institute for Statistics, Probability, Stochastic Operations Research and their Applications (EURANDOM), Eindhoven, Netherlands, March 7, 2016.

R. Soares Dos Santos, Random walk on random walks, University College London, Department of Mathematics, London, UK, June 15, 2016.

R. Soares Dos Santos, Random walk on random walks, Rhein-Main Kolloquium Stochastik, Johannes-Gutenberg Universität, Institut für Mathematik, Mainz, May 13, 2016.

W. van Zuijlen, Mean field Gibbs-non-Gibbs transitions, 6th Berlin--Oxford Meeting, December 8 - 10, 2016, University of Oxford, Mathematics Department, UK, December 9, 2016.

A. Cipriani, Extremes of some Gaussian random interfaces, Seminar Series in Probability and Statistics, Delft University of Technology, Department of Applied Probability, Netherlands, January 21, 2016.

A. Cipriani, The membrane model, seminar, Delft University of Technology, Department of Applied Probability, Netherlands, September 5, 2016.

A. Cipriani, The membrane model, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, September 29, 2016.

F. Flegel, Spectral localization in the random conductance model, 2nd Berlin Dresden Prague Würzburg Workshop on Mathematics of Continuum Mechanics, Technische Universität Dresden, Fachbereich Mathematik, December 5, 2016.

F. Flegel, Spectral localization vs. homogenization in the random conductance model, Summer School 2016, August 21 - 26, 2016, Research Training Group (RTG) 1845 ``Stochastic Analysis with Applications in Biology, Finance and Physics'', Hejnice, Czech Republic, August 22, 2016.

F. Flegel, Spectral localization vs. homogenization in the random conductance model, Probability Seminar at UCLA, University of California, Los Angeles, Department of Mathematics, Los Angeles, USA, October 13, 2016.

O. Gün, Fixation times for the mutation-selection model on random fitness landscapes, Joint Meeting of the SPP 1590 and 1819, September 28 - 29, 2016, Universität zu Köln, Köln, September 29, 2016.

W. König, A variational formula for the free energy of an interacting many-body system, Workshop ``Variational Structures and Large Deviations for Interacting Particle Systems and Partial Differential Equations'', March 15 - 18, 2016, European Institute for Statistics, Probability, Stochastic Operations Research and their Applications (EURANDOM), Eindhoven, Netherlands, March 17, 2016.

W. König, Connection times in large ad-hoc mobile networks, Workshop on Dynamical Networks and Network Dynamics, January 18 - 21, 2016, International Centre for Mathematical Science, Edinburgh, UK, January 18, 2016.

W. König, The mean-field polaron model, Workshop on Stochastic Processes in honour of Erwin Bolthausen's 70th birthday, September 14 - 16, 2016, Universität Zürich, Institut für Mathematik, Switzerland, September 15, 2016.

W. König, The spatially discrete parabolic Anderson model with time-dependent potential, ``Guided Tour: Random Media'' --- Special occasion to celebrate the 60th birthday of Frank den Hollander, December 14 - 16, 2016, EURANDOM, Eindhoven, Netherlands, December 16, 2016.

R.I.A. Patterson, Monte Carlo simulation of soot, King Abdullah University of Science and Technology (KAUST), Clean Combustion Research Center, Thuwal, Saudi Arabia, January 11, 2016.

R.I.A. Patterson, Pathwise LDPs for chemical reaction networks, 12th German Probability and Statistics Days 2016 --- Bochumer Stochastik-Tage, February 29 - March 4, 2016, Ruhr-Universität Bochum, Fakultät für Mathematik, March 4, 2016.

R.I.A. Patterson, Population balance simulation, University of Cambridge, Department for Chemical Engineering and Biotechnology, UK, May 5, 2016.

R.I.A. Patterson, Simulations of flame generated particles, Advances in Uncertainty Quantification Methods, Algorithms and Applications (UQAW 2016), January 5 - 10, 2016, King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia, January 5, 2016.

D.R.M. Renger, Large deviations for reacting particle systems: The empirical and ensemble processes, Workshop ``Interplay of Analysis and Probability in Applied Mathematics'', July 26 - August 1, 2015, Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach, July 30, 2015.

A. González Casanova Soberón, An individual-based model for the Lenski experiment, and the deceleration of the relative fitness, Workshop on Probabilistic Models in Biology, October 24 - 30, 2015, Playa del Carmen, Mexico, October 28, 2015.

A. González Casanova Soberón, Modeling the Lenski experiment, Genealogies in Evolution: Looking Backward and Forward, Workshop of the Priority Program (SPP) 1590 ``Probabilistic Structures in Evolution'', October 5 - 6, 2015, Goethe-Universität Frankfurt, October 6, 2015.

A. González Casanova Soberón, Modeling the Lenski experiment, 11. Doktorandentreffen Stochastik, Humboldt Universität zu Berlin, Institut für Mathematik (gemeinsam mit der TU Berlin), Berlin, September 2, 2015.

CH. Hirsch, From heavy-tailed Boolean models to scale-free Gilbert graphs, 18. Workshop on Stochastic Geometry, Stereology and Image Analysis, March 22 - 27, 2015, Universität Osnabrück, Lingen, March 23, 2015.

CH. Hirsch, Asymptotic properties of collective-rearrangement algorithms, International Conference on Geometry and Physics of Spatial Random Systems, September 6 - 11, 2015, Karlsruher Institut für Technology (KIT), Bad Herrenalb, September 7, 2015.

CH. Hirsch, Large-deviation principles in SINR-based wireless network models, Simons Conference on Networks and Stochastic Geometry, May 18 - 21, 2015, University of Texas, Austin, USA, May 18, 2015.

B. Jahnel, Classes of nonergodic interacting particle systems with unique invariant measure, Kyoto University, Research Institute for Mathematical Sciences, Kyoto, Japan, November 16, 2015.

B. Jahnel, Classes of non-ergodic interacting particle systems with unique invariant measure, Kac-Seminar, April 30 - May 3, 2015, Utrecht University, Department of Mathematics, Netherlands, May 1, 2015.

B. Jahnel, Classes of nonergodic interacting particle systems with unique invariant measure, Workshop ``Recent Trends in Stochastic Analysis and Related Topics'', September 20 - 21, 2015, Universität Hamburg, September 21, 2015.

B. Jahnel, Classes of non-ergodic interacting particle systems with unique invariant measure, Workshop ``Interacting Particles Systems and Non-Equilibrium Dynamics'', Institut Henri Poincaré, Paris, France, March 9 - 13, 2015.

P. Keeler, Large-deviation theory and coverage in mobile phone networks, Seminar ``Applied Probability'', The University of Melbourne, Department of Mathematics and Statistics, Australia, August 17, 2015.

P. Keeler, The Poisson--Dirichlet process and coverage in mobile phone networks, Stochastic Processes and Special Functions Workshop, August 13 - 14, 2015, The University of Melbourne, Melbourne, Australia, August 14, 2015.

P. Keeler, When do wireless network signals appear Poisson?, Simons Conference on Networks and Stochastic Geometry, May 18 - 21, 2015, University of Texas, Austin, USA, May 20, 2015.

P. Keeler, When do wireless network signals appear Poisson?, 18th Workshop on Stochastic Geometry, Stereology and Image Analysis, March 22 - 27, 2015, Universität Osnabrück, Lingen, March 24, 2015.

M. Maurelli, A large deviation principle for enhanced Brownian empirical measure, 4th Annual ERC Berlin-Oxford Young Researchers Meeting on Applied Stochastic Analysis, December 7 - 9, 2015, WIAS Berlin, December 8, 2015.

M. Maurelli, A large deviation principle for interacting particle SDEs via rough paths, 38th Conference on Stochastic Processes and their Applications, July 13 - 17, 2015, University of Oxford, Oxford-Man Institute of Quantitative Finance, UK, July 14, 2015.

M. Maurelli, Enhanced Sanov theorem for Brownian rough paths and an application to interacting particles, Seminar Stochastic Analysis, Imperial College London, UK, October 20, 2015.

M. Maurelli, Stochastic 2D Euler equations: A poorly correlated multiplicative noise regularizes the two-point motion, Universität Augsburg, Institut für Mathematik, March 24, 2015.

CH. Mukherjee, Compactness, large deviations and the mean-field polaron problem, Workshop ``Interplay of Analysis and Probability in Applied Mathematics'', July 27 - 31, 2015, Mathematisches Forschungsinstitut Oberwolfach, July 28, 2015.

D.R.M. Renger, The empirical process of reacting particles: Large deviations and thermodynamic principles, Minisymposium ``Real World Phenomena Explained by Microscopic Particle Models'' of the 8th International Congress on Industrial and Applied Mathematics (ICIAM 2015), August 8 - 22, 2015, International Council for Industrial and Applied Mathematics, Beijing, China, August 10, 2015.

R. Soares Dos Santos, Mass concentration in the parabolic Anderson model, Oberseminar Stochastik, Johannes-Gutenberg-Universität, Institut für Mathematik, Mainz, November 17, 2015.

R. Soares Dos Santos, Random walk on a dynamic random environment consisting of a system of independent simple symmetric random walks, Oberseminar Stochastik, Technische Universität Darmstadt, Fachbereich Mathematik, Darmstadt, January 22, 2015.

R. Soares Dos Santos, Random walk on random walks, YEP XII: Workshop on Random Walk in Random Environment, March 23 - 27, 2015, Technical University of Eindhoven, EURANDOM, Netherlands, March 24, 2015.

R. Soares Dos Santos, Random walk on random walks, Mathematical Physics Seminar, Université de Genève, Section de Mathématiques, Genève, Switzerland, April 27, 2015.

A. Cipriani, Extremes of the super critical Gaussian free field, Workshop ``Women in Probability 2015'', July 3 - 4, 2015, Technische Universität München, July 3, 2015.

A. Cipriani, Extremes of the supercritical Gaussian free field, Seminar Series in Probability and Statistics, Technical University of Delft, Applied Mathematics, Netherlands, June 11, 2015.

A. Cipriani, Extremes of the supercritical Gaussian free field, Probability Seminar, Leiden University, Netherlands, June 18, 2015.

A. Cipriani, Rates of convergence for extremes of geometric random variables and marked point processes, Workshop ``Interplay of Analysis and Probability in Applied Mathematics'', July 26 - August 1, 2015, Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach, July 28, 2015.

A. Cipriani, Rates of convergence for extremes of geometric random variables and marked point processes, Università degli Studi di Milano-Bicocca, Dipartimento di Matematica Applicazioni, Milano, Italy, March 30, 2015.

F. Flegel, Localization of the first Dirichlet-eigenvector in the heavy-tailed random conductance model, Summer School 2015 of the RTG 1845 Berlin-Potsdam ``Stochastic Analysis with Applications in Biology, Finance and Physics'', September 28 - October 3, 2015, Levico Terme, Italy, October 1, 2015.

F. Flegel, Localization of the first Dirichlet-eigenvector in the heavy-tailed random conductance model, Workshop ``Interplay of Analysis and Probability in Applied Mathematics'', July 26 - August 1, 2015, Mathematisches Forschungsinstitut Oberwolfach, July 30, 2015.

O. Gün, Branching random walks in random environments on hypercubes, Workshop on Random Walk in Random Environment, March 22 - 27, 2015, European Institute for Statistics, Probability, Stochastic Operations Research and their Applications (EURANDOM), Eindhoven, Netherlands, March 27, 2015.

O. Gün, Fluid and diffusion limits for the Poisson encounter-mating model, Mini-Workshop on Population Dynamics, April 6 - 17, 2015, Bŏgaziçi University Istanbul, Department of Mathematics, Istanbul, Turkey, April 6, 2015.

O. Gün, Stochastic encounter-mating model, Mathematical Model in Ecology and Evolution (MMEE 2015), July 7 - 13, 2015, Collège de France, Paris, France, July 8, 2015.

W. König, Moment asymptotics for a branching random walk in random environment, Applied Mathematics Seminars, University of Warwick, Mathematics Institute, Coventry, UK, November 6, 2015.

R.I.A. Patterson, Approximation errors for Smoluchowski simulations, 10 th IMACS Seminar on Monte Carlo Methods, July 6 - 10, 2015, Johannes Kepler Universität Linz, Austria, July 7, 2015.

R.I.A. Patterson, Particle systems, kinetic equations and their simulation, 8th International Congress on Industrial and Applied Mathematics, ICIAM 2015, August 8 - 15, 2015, CNCC - China National Convention Center, Beijing, China.

W. Wagner, Probabilistic models for the Schrödinger equation, 6th Workshop ``Theory and Numerics of Kinetic Equations'', June 1 - 4, 2015, Universität Saarbrücken, June 2, 2015.

CH. Hirsch, From heavy-tailed Boolean models to scale-free Gilbert graphs, Karlsruher Institut für Technologie, Institut für Stochastik, November 14, 2014.

A. Cipriani, Thick points for generalized Gaussian fields with different cut-offs, Berlin-Padova Young Researchers Meeting, October 23 - 25, 2014, Technische Universität Berlin, October 24, 2014.

A. Cipriani, Thick points for massive Gaussian free fields on R^d, School and Workshop on Random Interacting Systems, June 23 - 27, 2014, University of Bath, UK, June 23, 2014.

O. Gün, Stochastic encounter-mating models, University of Leiden, Mathematical Institute, Netherlands, September 4, 2014.

W. König, A variational formula for the free energy of a many-Boson system, IKERBASQUE, Basque Foundation for Science, Bilbao, Spain, May 29, 2014.

W. König, Von der Binomialverteilung zur Normalverteilung, 11th German Probability and Statistics Days 2014, March 4 - 7, 2014, Ulm, March 6, 2014.

R.I.A. Patterson, Statistical error analysis for coagulation-advection simulations, Eleventh International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing (MCQMC 2014), April 6 - 11, 2014, KU Leuven, Belgium, April 8, 2014.

R.I.A. Patterson, Statistical error analysis for coagulation-advection simulations, University of Cambridge, Department of Chemical Engineering and Biotechnology, UK, May 1, 2014.

R.I.A. Patterson, Stochastic numerical methods for coagulating particles, Seminar ``Geophysical Fluid Dynamics'', Freie Universität Berlin, Institut für Mathematik, June 4, 2014.

W. Wagner, Heat generation in the electrothermal Monte Carlo method, The 18th European Conference on Mathematics for Industry 2014 (ECMI 2014), Minisymposium ``Semiclassical and Quantum Transport in Semiconductors and Low Dimensional Materials'', June 9 - 13, 2014, Taormina, Italy, June 11, 2014.

W. Wagner, Random cloud models for the Schrödinger equation, Sapienza -- Università di Roma, Dipartimento di Matematica, Italy, October 9, 2014.

L. Avena, A local CLT for some convolution equations with applications to self-avoiding walks, Università degli Studi di Roma ``La Sapienza'', Dipartimento di Matematica, Italy, December 17, 2013.

G. Faraud, Connection times in large ad-hoc networks, Ecole de Printemps ``Marches Aléatoires, Milieux Aléatoires, Renforcements'' (MEMEMO2), June 10 - 14, 2013, Aussois, France, June 13, 2013.

O. Gün, Aging for GREM-like trap models, CIRM-Conference "`Dynamical and Disordered Systems"', February 11 - 15, 2013, Centre International de Rencontres Mathématiques, Marseille, France, February 15, 2013.

O. Gün, Moment asymptotics for branching random walks in random environment, Workshop on Disordered Systems, June 24 - 28, 2013, Centre International de Rencontres Mathématiques, Marseille, France, June 24, 2013.

W. König, A variational formula for the free energy of a many-boson system, Random Combinatorial Structures and Statistical Mechanics, May 6 - 10, 2013, University of Warwick, Mathematics Institute, Warwick in Venice, Palazzo Pesaro-Papafava, Italy, May 10, 2013.

W. König, Geordnete Irrfahrten, Technische Universität Darmstadt, Fachbereich Mathematik, October 23, 2013.

W. König, Large deviations for the local times of a random walk among random conductances, Random Walks: Crossroads and Perspectives, Satellite Meeting of the Erdős Centennial Conference, June 24 - 28, 2013, Alfréd Rényi Institute of Mathematics, Budapest, Hungary, June 28, 2013.

W. König, Upper tails of self-intersection local times: Survey of proof techniques, 12. Erlanger-Münchner Tag der Stochastik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Department Mathematik, July 12, 2013.

W. König, Upper tails of self-intersection local times: Survey of proof techniques, Kyoto University, Research Institute for Mathematical Sciences, Japan, April 12, 2013.

R.I.A. Patterson, Monte Carlo simulation of nano-particle formation, University of Technology Eindhoven, Institute for Complex Molecular Systems, Netherlands, September 5, 2013.

R.I.A. Patterson, Stochastic methods for particle coagulation problems in flows with boundaries, 5th Workshop ``Theory and Numerics of Kinetic Equations'', May 13 - 15, 2013, Universität des Saarlandes, Saarbrücken, May 14, 2013.

W. Wagner, Stochastic particle methods for population balance equations, 5th Workshop ``Theory and Numerics of Kinetic Equations'', May 13 - 15, 2013, Universität des Saarlandes, Saarbrücken, May 13, 2013.

T. Wolff, Annealed asymptotics for occupation time measures of a random walk among random conductances, ``Young European Probabilists 2013 (YEP X)'' and ``School on Random Polymers'', January 8 - 12, 2013, EURANDOM, Eindhoven, Netherlands, January 10, 2013.

S. Jansen, Cluster size distributions and approximate random partition model for Gibbs measures in continuous configuration space, 10th German Probability and Statistics Days (GPSD2012), March 6 - 9, 2012, Johannes Gutenberg-Universität Mainz, Mainz, March 8, 2012.

S. Jansen, Cluster size distributions at low temperature and low density, Interplay of Analysis and Probability in Physics, January 22 - 28, 2012, Mathematisches Forschungsinstitut Oberwolfach, January 24, 2012.

S. Jansen, Fermionic and bosonic Laughlin state on thick cylinders, Mathematical Physics Seminar, University of Alabama, Department of Mathematics, Birmingham, USA, February 17, 2012.

S. Jansen, Random partitions and heavy-tailed variables in statistical mechanics, Probability Seminar, University of Alabama, Department of Mathematics, Birmingham, USA, February 9, 2012.

O. Gün, Moment asymptotics for branching random walks in random environment, Probability Laboratory at Bath, Prob-L@b Seminar, University of Bath, Department of Mathematical Sciences, UK, August 20, 2012.

W. König, Large deviations for cluster size distributions in a classical many-body system, Probability Laboratory at Bath, Prob-L@b Seminar, University of Bath, Department of Mathematical Sciences, UK, August 20, 2012.

W. König, Large deviations for the cluster size distribution in a classical interacting many-particle system, Warwick Mathematics Institute Seminars, University of Warwick, Mathematics Institute, Coventry, UK, August 1, 2012.

O. Gün, Dynamics of GREM-like trap models, Interplay of Analysis and Probability in Physics, January 22 - 28, 2012, Mathematisches Forschungsinstitut Oberwolfach, January 27, 2012.

O. Gün, Multilevel trap models and aging for spin glasses, Seminar Wahrscheinlichkeitstheorie, Universität Wien, Fakultät für Mathematik, Austria, May 21, 2012.

W. König, Eigenvalue order statistics and mass concentration in the parabolic Anderson model, SFB/TR12 Workshop, November 4 - 8, 2012, Universität zu Köln, SFB TR12 ``Symmetries and Universality in Mesoscopic Systems'', Langeoog, November 7, 2012.

W. König, Large deviations for cluster size distributions in a classical many-body system, Oberseminar Stochastik, Rheinische Friedrich-Wilhelms-Universität Bonn, Institut für Angewandte Mathematik, January 12, 2012.

W. König, Large deviations for the cluster size distributions in a classical interacting many-particle system with Lennard--Jones potential, Mark Kac Seminar, Eindhoven University of Technology, Netherlands, November 9, 2012.

W. König, Large deviations for the local times of random walk among random conductances, EPSRC Symposium Workshop -- Large Scale Behaviour of Random Spatial Models, May 28 - June 1, 2012, University of Warwick, Mathematical Institute, Coventry, UK, June 1, 2012.

W. König, Moment asymptotics for branching random walks in random environment, Stochastik-Oberseminar, Westfälische Wilhelms-Universität Münster, Institut für Mathematische Statistik, December 6, 2012.

W. König, Ordered random walks, Stochastisches Kolloquium, Georg-August-Universität Göttingen, Institut für Mathematische Stochastik, June 6, 2012.

R.I.A. Patterson, Convergence of simulable processes for coagulation with transport, WIAS-workshop "From particle systems to dierential equations", February 21 - 23, 2012, WIAS Berlin, February 22, 2012.

R.I.A. Patterson, Soot as a boundary value problem, University of Cambridge, Department of Chemical Engineering, UK, May 8, 2012.

R.I.A. Patterson, Stochastic methods for particle populations in flows, 3rd European Seminar on Computing, June 25 - 29, 2012, Pilsen, Czech Republic.

W. Wagner, Coagulation equations and particle systems, WIAS-workshop "From Particle Systems to Differential Equations", February 21 - 23, 2012, WIAS Berlin, February 22, 2012.

W. Wagner, Stochastic particle methods for coagulation problems, 28th International Symposium on Rarefied Gas Dynamics, July 9 - 13, 2012, University of Zaragoza, Spain, July 9, 2012.

T. Wolff, Annealed asymptotics for occupation time measures of a random walk among random conductances, University of California at Los Angeles, Mathematics Department, USA, October 24, 2012.

T. Wolff, Non-exit probability from a time-dependent region of a random walk among random conductances, Interplay of Analysis and Probability in Physics, January 22 - 28, 2012, Mathematisches Forschungsinstitut Oberwolfach, January 26, 2012.

S. Jansen, Cluster size distributions at low density and low temperature, The University of Arizona, Department of Mathematics, Tucson, USA, April 13, 2011.

S. Jansen, Fermionic and bosonic Laughlin state on thick cylinders, Venice 2011 --- Quantissima in the Serenissima, August 1 - 5, 2011, University of Warwick (VB), Warwick in Venice, Italy, August 4, 2011.

S. Jansen, Large deviations for interacting many-particle systems in the Saha regime, Berlin-Leipzig Seminar on Analysis and Probability Theory, July 8, 2011, Technische Universität Clausthal, Institut für Mathematik, July 8, 2011.

S. Jansen, Random partitions in statistical physics, 5th International Conference on Stochastic Analysis and its Applications, September 5 - 9, 2011, Hausdorff Center for Mathematics and Rheinische Friedrich-Wilhelms-Universität Bonn, September 8, 2011.

S. Jansen, Statistical mechanics at low density and low temperature: Cross-over transitions from small to large cluster sizes, 2011 School on Mathematical Statistical Physics, August 29 - September 4, 2011, Charles University of Prague, Center for Theoretical Study, and Academy of Sciences of the Czech Republic, Institute of Theoretical Computer Science, September 8, 2011.

S. Jansen, Random partitions in statistical physics, Warwick Statistical Mechanics Seminar, University of Warwick, Department of Mathematics, UK, November 17, 2011.

M. Roberts, The unscaled paths of branching Brownian motion, Oberseminar "Biological Models and Statistical Mechanics", The University of Nottingham, School of Community Health Sciences, UK, January 17, 2011.

G. Faraud, Marche aléatoires en milieu aléatoire: Le cas des arbres, École Normale Supérieure de Lyon, Unité de Mathématiques Pures et Appliquées, Séminaire de Probabilités, France, February 3, 2011.

G. Faraud, Random walks in random environment on trees, 2011 School on Mathematical Statistical Physics, August 28 - September 9, 2011, Charles University of Prague, Center for Theoretical Study, and Academy of Sciences of the Czech Republic, Institute of Theoretical Computer Science, September 2, 2011.

W. König, A variational formula for the free energy of a many-Boson system, University of California at Los Angeles, Department of Mathematics, USA, April 11, 2011.

W. König, Eigenvalue order statistics and mass concentration in the parabolic Anderson model, Berlin-Leipzig Seminar on Analysis and Probability Theory, Technische Universität Clausthal, Institut für Mathematik, July 8, 2011.

W. König, Eigenvalue order statistics for the heat equation with random potential, Extreme Value Statistics in Mathematics, Physics and Beyond, July 4 - 8, 2011, Lorentz Center, International Center for Workshops in the Sciences, Leiden, Netherlands, July 6, 2011.

W. König, Large deviations for cluster size distributions in a classical many-body system, Università Ca' Foscari Venezia, Dipartimento di Management, Italy, October 13, 2011.

W. König, Localisation of the parabolic Anderson model in one island, Jahrestagung der Deutschen Mathematiker-Vereinigung (DMV) 2011, September 20 - 22, 2011, Universität zu Köln, Mathematisches Institut, September 20, 2011.

W. König, Ordered random walks, University of California at Los Angeles, Department of Mathematics, USA, April 4, 2011.

W. König, Phase transitions for a dilute particle system with Lennard--Jones potential, Ludwig-Maximilians-Universität München, Mathematisches Institut, January 20, 2011.

W. König, The parabolic Anderson model, 2011 School on Mathematical Statistical Physics, September 4 - 9, 2011, Charles University of Prague, Center for Theoretical Study, and Academy of Sciences of the Czech Republic, Institute of Theoretical Computer Science.

W. König, The universality classes in the parabolic Anderson model, Technische Universität Dresden, Institut für Analysis, June 24, 2011.

W. König, Upper tails of self-intersection local times: Survey of proof techniques, University of Warwick, Mathematics Institute, Coventry, UK, February 17, 2011.

T. Wolff, Annealed behaviour of local times in the random conductance model, 2011 School on Mathematical Statistical Physics, September 4 - 9, 2011, Charles University of Prague, Center for Theoretical Study, and Academy of Sciences of the Czech Republic, Institute of Theoretical Computer Science, September 8, 2011.

T. Wolff, Random walk among random conductances, Spring Meeting Beijing/Bielefeld --- Berlin/Zurich of the International Research Group Stochastic Models of Complex Processes, March 30 - April 1, 2011, Technische Universität Berlin, March 31, 2011.

T. Wolff, The parabolic Anderson model from the perspective of a moving catalyst, 7th Cornell Probability Summer School, July 8 - 24, 2011, Cornell University, Ithaka, USA, July 18, 2011.

G. Faraud, Marche aléatoires en milieu aléatoire: Le cas des arbres, Université Paris VI ``Pierre et Marie Curie'', Laboratoire de Probabilités et Modèles Aléatoires, France, November 21, 2011.

O. Gün, Parabolic Anderson model with finite number of moving catalysts, Istanbul Center For Mathematical Sciences, Turkey, December 23, 2011.

O. Gün, Trap models and aging for spin glasses, Bogazici University, Department of Mathematics, Istanbul, Turkey, December 21, 2011.

O. Gün, Trap models and aging for spin glasses, Koc University, Istanbul, Turkey, December 27, 2011.

W. König, Eigenvalue order statistics and mass concentration in the parabolic Anderson model, Technische Universität München, Fakultät für Mathematik, December 21, 2011.

W. König, Eigenvalue order statistics and mass concentration in the parabolic Anderson model, Tokyo Institute of Technology, Department of Mathematics, Japan, December 9, 2011.

W. König, Large deviations for cluster size distributions in a classical many-body system, 10th Workshop ``Stochastic Analysis on Large Scale Interacting Systems'', December 5 - 7, 2011, Kochi University, Faculty of Science, Shikoku, Japan, December 6, 2011.

W. König, Large deviations for cluster size distributions in a classical many-body system, Universität Augsburg, Institut für Mathematik, December 22, 2011.

R.I.A. Patterson, Simulating coagulating particles in flows, University of Cambridge, Department of Chemical Engineering, UK, October 17, 2011.

R.I.A. Patterson, Simulating coagulating particles with advection, University of Cambridge, Department of Chemical Engineering, UK, May 3, 2011.

W. Wagner, Direct simulation Monte Carlo algorithms, Università di Catania, Dipartimento di Matematica e Informatica, Italy, October 6, 2011.

W. Wagner, Stochastic particle methods, 8th International Conference on Large-Scale Scientific Computations, June 6 - 10, 2011, Institute of Information and Communication Technologies, Bulgarian Academy of Sciences and Society for Industrial and Applied Mathematics (SIAM), Sozopol, Bulgaria, June 6, 2011.

S. Jansen, M. Aizenman, P. Jung, Symmetry breaking in quasi 1D Coulomb systems, Combinatorics and Analysis in Spatial Probability --- ESF Mathematics Conference in Partnership with EMS and ERCOM, Eindhoven, Netherlands, December 12 - 18, 2010.

S. Jansen, Combinatorics and Analysis in Spatial Probability, ESF Mathematics Conference in partnership with EMS and ERCOM:, December 12 - 18, 2010, EURANDOM, Eindhoven, Netherlands.

B. Metzger, The parabolic Anderson model: The asymptotics of the statistical moments and Lifshitz tails revisited, EURANDOM, Eindhoven, Netherlands, December 1, 2010.

M. Sander, R.I.A. Patterson, A. Braumann, A. Rai, M. Kraft, Boundary value stochastic particle methods for population balance problems, 4th International Conference on Population Balance Modelling (PBM 2010), Berlin, September 15 - 17, 2010.

W. König, Die Universalitätsklassen im parabolischen Anderson-Modell, Mathematisches Kolloquium, Technische Universität Darmstadt, Fachbereich Mathematik, July 7, 2010.

W. König, On random matrix theory, Introductory Course for the IRTG Summer School Pro*Doc/IRTG Berlin-Zürich ``Stochastic Models of Complex Processes'' (Disentis, Switzerland, July 26--30, 2010), July 21 - 22, 2010, Technische Universität Berlin, July 21, 2010.

W. König, Ordered random walks, Augsburger Mathematisches Kolloquium, Universität Augsburg, Institut für Mathematik, January 26, 2010.

W. König, Ordered random walks, Mathematisches Kolloquium der Universität Trier, Fachbereich Mathematik, April 29, 2010.

W. König, Phase transitions for dilute particle systems with Lennard--Jones potential, University of Bath, Department of Mathematical Sciences, UK, April 14, 2010.

W. König, Phase transitions for dilute particle systems with Lennard--Jones potential, Università di Roma ``Tor Vergata'', Dipartimento di Matematica, Italy, November 17, 2010.

W. König, The parabolic Anderson model, XIV Escola Brasileira de Probabilidade, August 2 - 7, 2010, Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Brazil.

W. König, Upper tails of self-intersection local times of random walks: Survey of proof techniques, Excess Self-Intersection Local Times and Related Topics, December 6 - 10, 2010, Université de Marseille, Centre International de Rencontres Mathématiques (CIRM), France, December 7, 2010.

W. Wagner, Kinetic equations and Markov jump processes, Isaac Newton Institute for Mathematical Sciences, Programme: Partial Differential Equations in Kinetic Theories, Cambridge, UK, November 29, 2010.

W. König, Modeling and understanding random Hamiltonians: Beyond monotonicity, linearity and independence, Miniworkshop ``Numerics for Kinetic Equations'', December 6 - 12, 2009, Mathematisches Forschungsinstitut Oberwolfach.

W. König, Phase transitions for dilute particle systems with Lennard--Jones potential, Workshop on Mathematics of Phase Transitions: Past, Present, Future, November 12 - 15, 2009, University of Warwick, Coventry, UK, November 15, 2009.

W. Wagner, Explosion properties of random fragmentation models, Workshop ``Coagulation et Fragmentation Stochastiques'', April 15 - 18, 2008, Université Paris VI, Laboratoire de Probabilités, France, April 16, 2008.

W. Wagner, Explosion properties of random fragmentation models, Università di Catania, Dipartimento di Matematica e Informatica, Italy, May 8, 2008.

W. Wagner, Introduction to Markov jump processes, Università di Catania, Dipartimento di Matematica e Informatica, Italy, May 7, 2008.

A. Weiss, Escaping the Brownian stalkers, BRG Workshop on Stochastic Models from Biology and Physics, October 9 - 10, 2006, Johann Wolfgang Goethe-Universität Frankfurt, October 10, 2006.

A. Weiss, Escaping the Brownian stalkers, 5th Prague Summer School 2006 "`Statistical Mathematical Mechanics"', September 10 - 23, 2006, Charles University, Center for Theoretical Study and Institute of Theoretical Computer Science, Prague, Czech Republic, September 20, 2006.

W. Wagner, Explosion phenomena in stochastic coagulation-fragmentation model, October 17 - 24, 2006, Università di Catania, Dipartimento di Matematica e Informatica, Italy, October 19, 2006.

W. Wagner, Explosion phenomena in stochastic coagulation-fragmentation models, University of Cambridge, Centre for Mathematical Sciences, UK, May 9, 2006.

W. Wagner, Gelation in stochastic models, Workshop ``Stochastic Methods in Coagulation and Fragmentation'', December 8 - 12, 2003, Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, December 10, 2003.

V. Betz, H. Schäfer, L. Taggi, Interacting self-avoiding polygons, Preprint no. arXiv:1902.08517, Cornell University Library, 2019.
Abstract
We consider a system of self-avoiding polygons interacting through a potential that penalizes or rewards the number of mutual touchings and we provide an exact computation of the critical curve separating a regime of long polygons from a regime of localized polygons. Moreover, we prove the existence of a sub-region of the phase diagram where the self-avoiding polygons are space filling and we provide a non-trivial characterization of the regime where the polygon length admits uniformly bounded exponential moments.

D. Heydecker , R.I.A. Patterson, Kac interaction clusters: A bilinear coagulation equation and phase transition, Preprint no. arXiv:1902.07686, Cornell University Library, 2019.
Abstract
We consider the interaction clusters for Kac's model of a gas with quadratic interaction rates, and show that they behave as coagulating particles with a bilinear coagulation kernel. In the large particle number limit the distribution of the interaction cluster sizes is shown to follow an equation of Smoluchowski type. Using a coupling to random graphs, we analyse the limiting equation, showing well-posedness, and a closed form for the time of the gelation phase transition tg when a macroscopic cluster suddenly emerges. We further prove that the second moment of the cluster size distribution diverges exactly at tg. Our methods apply immediately to coagulating particle systems with other bilinear coagulation kernels.

A.D. Mcguire, S. Mosbach, G. Reynolds, R.I.A. Patterson, E.J. Bringley, N.A. Eaves, J. Dreyer, M. Kraft, Analysing the effect of screw configuration using a stochastic twin-screw granulation model, Technical report no. 195, University of Cambridge, c4e-Preprint Series, 2018.

D.R.M. Renger, P. Koltai , From large deviations to transport semidistances: Coherence analysis for finite Lagrangian data, Preprint no. arXiv:1709.02352, Cornell University Library, arXiv.org, 2017.
Abstract
Onsager's 1931 `reciprocity relations' result connects microscopic time-reversibility with a symmetry property of corresponding macroscopic evolution equations. Among the many consequences is a variational characterization of the macroscopic evolution equation as a gradient-flow, steepest-ascent, or maximal-entropy-production equation. Onsager's original theorem is limited to close-to-equilibrium situations, with a Gaussian invariant measure and a linear macroscopic evolution. In this paper we generalize this result beyond these limitations, and show how the microscopic time-reversibility leads to natural generalized symmetry conditions, which take the form of generalized gradient flows.

L. Andreis, A. Asselah, P. Dai Pra , Ergodicity of a system of interacting random walks with asymmetric interaction, Preprint no. arXiv:1702.02754, Cornell University Library, arXiv.org, 2017.
Abstract
We study N interacting random walks on the positive integers. Each particle has drift delta towards infinity, a reflection at the origin, and a drift towards particles with lower positions. This inhomogeneous mean field system is shown to be ergodic only when the interaction is strong enough. We focus on this latter regime, and point out the effect of piles of particles, a phenomenon absent in models of interacting diffusion in continuous space.

L. Andreis, P. Dai Pra, M. Fischer, McKean--Vlasov limit for interacting systems with simultaneous jumps, Preprint no. arXiv:1704.01052, Cornell University Library, arXiv.org, 2017.
Abstract
Motivated by several applications, including neuronal models, we consider the McKean-Vlasov limit for mean-field systems of interacting diffusions with simultaneous jumps. We prove propagation of chaos via a coupling technique that involves an intermediate process and that gives a rate of convergence for the W1 Wasserstein distance between the empirical measures of the two systems on the space of trajectories D([0,T],R^d).

L. Andreis, F. Polito, L. Sacerdote, On a class of time-fractional continuous-state branching processes, Preprint no. arXiv:1702.03188, Cornell University Library, arXiv.org, 2017.
Abstract
We propose a class of non-Markov population models with continu- ous or discrete state space via a limiting procedure involving sequences of rescaled and randomly time-changed Galton?Watson processes. The class includes as specific cases the classical continuous-state branching processes and Markov branching processes. Several results such as the expressions of moments and the branching inequality governing the evolution of the process are presented and commented. The gener- alized Feller branching diffusion and the fractional Yule process are analyzed in detail as special cases of the general model.

L. Andreis, D. Tovazzi, Coexistence of stable limit cycles in a generalized Curie--Weiss model with dissipation, Preprint no. arXiv:1711.05129, Cornell University Library, arXiv.org, 2017.
Abstract
In this paper, we modify the Langevin dynamics associated to the generalized Curie-Weiss model by introducing noisy and dissipative evolution in the interaction potential. We show that, when a zero-mean Gaussian is taken as single-site distribution, the dynamics in the thermodynamic limit can be described by a finite set of ODEs. Depending on the form of the interaction function, the system can have several phase transitions at different critical temperatures. Because of the dissipation effect, not only the magnetization of the systems displays a self-sustained periodic behavior at sufficiently low temperature, but, in certain regimes, any (finite) number of stable limit cycles can exist. We explore some of these peculiarities with explicit examples.

A. González Casanova Soberón, N. Kurt, A. Wakolbinger, L. Yuan, An individual-based model for the Lenski experiment, and the deceleration of the relative fitness, Preprint no. arxiv.org:1505.01751, Cornell University Library, arXiv.org, 2016.

J. Blath, A. González Casanova Soberón, N. Kurt, M. Wilke-Berenguer, A new coalescent for seed-bank models, Preprint no. arxiv.org:1411.4747, Cornell University Library, arXiv.org, 2016.

J.-D. Deuschel, P. Friz, M. Maurelli, M. Slowik, The enhanced Sanov theorem and propagation of chaos, Preprint no. arxiv:1602.08043, Cornell University Library, arXiv.org, 2016.

M. Kraft, W. Wagner, A numerical scheme for the Random Cloud Model, Technical report no. 173, c4e-Preprint Series, 2016.

K.F. Lee, M. Dosta, A.D. Mcguire, S. Mosbach, W. Wagner, S. Heinrich, M. Kraft, Development of a multi-compartment population balance model for high-shear wet granulation with Discrete Element Method, Technical report no. 170, c4e-Preprint Series, 2016.
Abstract
This paper presents a multi-compartment population balance model for wet granulation coupled with DEM (Discrete Element Method) simulations. Methodologies are developed to extract relevant data from the DEM simulations to inform the population balance model. First, compartmental residence times are calculated for the population balance model from DEM. Then, a suitable collision kernel is chosen for the population balance model based on particle-particle collision frequencies extracted from DEM. It is found t hat the population balance model is able to predict the trends exhibited by the experimental size and porosity distributions by utilising the information provided by the DEM simulations.

R.I.A. Patterson, S. Simonella, W. Wagner, Kinetic theory of cluster dynamics, Preprint no. arXiv: 1601.05838, Cornell University Libary, arXiv.org, 2016.
Abstract
In a Newtonian system with localized interactions the whole set of particles is naturally decomposed into dynamical clusters, dened as nite groups of particles having an in uence on each other's trajectory during a given interval of time. For an ideal gas with shortrange intermolecular force, we provide a description of the cluster size distribution in terms of the reduced Boltzmann density. In the simplied context of Maxwell molecules, we show that a macroscopic fraction of the gas forms a giant component in nite kinetic time. The critical index of this phase transition is in agreement with previous numerical results on the elastic billiard.

E.K.Y. Yapp, R.I.A. Patterson, J. Akroyd, S. Mosbach, E.M. Adkins, J.H. Miller, M. Kraft, Numerical simulation and parametric sensitivity study of optical band gap in a laminar co-flow ethylene diffusion flame, Technical report no. 159, University of Cambridge, c4e-Preprint Series, 2015.
Abstract
A detailed population balance model is used to perform a parametric sensitivity study on the computed optical band gap (OBG) of polycyclic aromatic hydrocarbons (PAHs) in a laminar co-flow ethylene diffusion flame. OBG may be correlated with the number of aromatic rings in PAHs which allows insights into which are the key species involved in the formation of soot. PAH size distributions are computed along the centerline and in the wings of the flame. We compare our simulations with experimentally determined soot volume fraction and OBG (derived from extinction measurements) from the literature. It is shown that the model predicts reasonably well the soot volume fraction and OBG throughout the flame. We find that the computed OBG is most sensitive to the size of the smallest PAH which is assumed to contribute to the OBG. The best results are obtained accounting for PAH contribution in both gas and particle phases assuming a minimum size of ovalene (10 rings). This suggests that the extinction measurements show a significant absorption by PAHs in the gas phase at the visible wavelength that is used, which has been demonstrated by experiments in the literature. It is further shown that PAH size distributions along the centerline and in the wings are unimodal at larger heights above burner. Despite the different soot particle histories and residence times in the flame, the PAH size associated with both modes are similar which is consistent with the near-constant OBG that is observed experimentally. The simulations indicate that the transition from the gas phase to soot particles begins with PAHs with as few as 16 aromatic rings, which is consistent with experimental observations reported in the literature.

A. Cipriani, S.H. Rajat, Thick points for Gaussian free fields with different cut-offs, Preprint no. arXiv:1407.5840, Cornell University Library, arXiv.org, 2014.

W.J. Menz, R.I.A. Patterson, W. Wagner, M. Kraft, Application of stochastic weighted algorithms to a multidimensional silica particle model, Preprint no. 120, University of Cambridge, Cambridge Center for Computational Chemical Engineering, 2012.

S. Shekar, A.J. Smith, M. Kraft, W. Wagner, On a multivariate population balance model to describe the structure and composition of silica nanoparticles, Technical report no. 105, c4e-Preprint Series, Cambridge, 2011.

A. Braumann, M. Kraft, W. Wagner, Numerical study of a stochastic particle algorithm solving a multidimensional population balance model for high shear granulation, Technical report no. 93, c4e-Preprint Series, Cambridge Centre for Computational Chemical Engineering, University of Cambridge, Department of Chemical Engineering, 2010.
Abstract
This paper is concerned with computational aspects of a multi-dimensional population balance model of a wet granulation process. Wet granulation is a manufacturing method to form composite particles, granules, from small particles and binders. A detailed numerical study of a stochastic particle algorithm for the solution of a five-dimensional population balance model for wet granulation is presented. Each particle consists of two types of solids (containing pores) and of external and internal liquid (located in the pores). Several transformations of particles are considered, including coalescence, compaction and breakage. A convergence study is performed with respect to the parameter that determines the number of numerical particles. Averaged properties of the system are computed. In addition, the ensemble is subdivided into practically relevant size classes and analysed with respect to the amount of mass and the particle porosity in each class. These results illustrate the importance of the multi-dimensional approach. Finally, the kinetic equation corresponding to the stochastic model is discussed.

W. Wagner, Explosion phenomena in stochastic coagulation-fragmentation models, Preprint no. NI04006-IGS, Isaac Newton Institute for Mathematical Sciences, 2004.